Hey all, I need help. In the game I am running currently I am having trouble making one of my players feel integral to the story. This is mostly because they are non-magical. I will quickly present in 3 paragraphs the game I am running, the characters in it, and a quick synopsis of where they are in the story so that you have a better understanding of my predicament. 

The game I run is set in a home-brewed campaign setting that I made, in a small kingdom that has a skewed perspective on magic. Their laws around magic are basically boiled down to this: if we can't understand it, we don't trust it. Powers and magic from gods are respected and trusted (especially since I have a small pantheon in my campaign where none of the gods are evil or good, just embodiments of different aspects of creation), whereas a person with inherent magical talents is sent to prison. However, magic that is studied is safe, since it is almost seen as a sort of pseudo-science. In simpler D&D terms, Paladins, Clerics, and Wizards are totally safe, but a Warlock or a Sorcerer is sent to prison to be tested, studied, and "cured" of their "curse." 

The party in this game is made up of six (later seven) players: a Beast Master Drow Ranger who hails from an outside country of slavers and raiders, a Human Paladin of Vengeance who serves the god of Revenge, a Lizardfolk Cleric of the Forge who hails from another country of dragon worshipers wishing to escape his country and their religion, a Half-orc Warlock of the Archfey who also recently multiclassed into a Sea Sorcerer who is a farm boy struggling to understand the powers placed upon him, a Human Wild Magic Sorcerer who is a jeweler from a country that runs on and loves magic, and the character this whole question is around: a Wood Elf Open Hand Monk who left her family tribe to see the world and experience all it has to offer. It is an AWESOME menagerie of characters, all of which have such amazing character arcs ahead of them... except for the monk. 

Here is the story: the party was hired by a dwarven historian to locate a lost underground military outpost of some long-ago war made by gnomes. This outpost was supposedly one of many, and the first lead this historian had found in decades. However, once the outpost was found, the party soon discovered that it was infested with monsters who had all been hired by another group vying for control of the run-down base. This was because this outpost had once been a forge for magical items to be used in this gnome war, and the group in question was hoping to reopen it again to create an army. The party, after defeating the monsters and their leader, found out that this leader was part of an organization (or, an Agency) that planned to overthrow the King of this country and remake all the laws so that magic users could live freely without fear that they would become lab experiments. The party didn't think much of it, as it seemed like they were a small group who wouldn't stand a chance against the kingdom. So they cleared out the military outpost and headed back home, where they went on a quest to avenge the murdered wife of the Paladin. However, afterward, they were hired by the dwarven historian again to try to restart the magical forge at the outpost and find the other outposts, rumored to be scattered across the countryside. So they returned to the original outpost, only to find a clan of goblins had made their home there. This is also where they met their newest party member: a Bugbear Druid of the Stars who follows the stars in the sky as their god and guide. 

Now, this all sounds amazing so far, right? A group of adventurers who have so many possible futures ahead of them. However, here is where we get to our problem. See, each of the characters has some magical being or power residing around or within them, pulling them towards a certain future. For the Cleric, Druid, and the Paladin, it is their god. For the Ranger, it is the very magical monsters she hunts to bring back to her country and create a massive monster army to be reckoned with. For the Sorcerer, their magic puts a target on their back in this kingdom and they are already struggling to understand how to best use and control their magic as it is. For the Warlock/Sorcerer, he has magical entities living inside his psyche who wish to use him as a puppet, forcing him to learn to control his power. But our Monk, our wonderful bubbly cheery happy-go-lucky monk, has no magical entities, has no powers, has nothing drawing them towards magic. I have spoken with the player of this monk and they have no desire to have anything magical with this character (except maybe magic items but, even then, not preferred). They want to keep her mundane, with her skills being her power. Now, for a better picture, here is a full rundown of this Monk: she is a clothing maker who wants to see the world, doesn't agree with her families morals and choices (hint: arranged marriage and such) and isn't afraid to break a few skulls if her friends are in danger. But there is a problem, her character stands out a LOT. Not only is she not magical, which causes her to be singled out in many situations, but her role in combat is also kind of.. stuck. The Sorcerer is a spell slinger, the Ranger is a marksman, the Cleric is a tank who soaks up damage, the Druid is a healer and support, the Warlock/Sorcerer is another spell slinger but also can handle themselves in melee because of their Pact of the Blade, and the Paladin is the main damage dealer and tank with his magic weapons (that he forged himself after many sessions of work as a blacksmith). The Monk has no purpose in combat except to catch up to running targets thanks to fast monk speed and consistent damage some of the time. And outside of combat, her character wants to travel and the story so far has kept the players stuck in this one country, making her even more stuck! 

In short, I have a character who is so unique to the party that it is hard to figure out how to write her into the story! On one hand, I could just get another magical being to take interest in her (maybe because of her want to see the world) so she doesn't feel left out of the many dream sequences my other PCs have. However, I worry that this will demean the value of having a completely non-magical party member and make my players think that you can't be important without being magical. 

What should I do? How do you write amazing arcs for characters with no ties to anything at all? I would greatly appreciate some help and guidance. 

No ties to magic doesn't mean no ties to anything at all. There are millions of stories that have no magic in them.

Is the monk part of an organization? What is its nature?

Do they have a family? What are its problems?  

What about their martial art technique? What's special about it? Does it have any secrets?

The player can help with ideas too: it's not all up to you. You have to consider, as well, what the player wants. They haven't given their character an aim beyond "travel." Maybe they're happier not being the centre of attention?

So your issue is how the monk has no magic?

OK, well let me just copy 'n' paste a snippet of the Monk intro and I'll emphasize something a little bit as you seemed to have missed it.

Whatever their discipline, monks are united in their ability to magically harness the energy that flows in their bodies. Whether channeled as a striking display of combat prowess or a subtler focus of defensive ability and speed, this energy infuses all that a monk does.

The Magic of Ki

Monks make careful study of a magical energy that most monastic traditions call ki. This energy is an element of the magic that suffuses the multiverse—specifically, the element that flows through living bodies. Monks harness this power within themselves to create magical effects and exceed their bodies’ physical capabilities, and some of their special attacks can hinder the flow of ki in their opponents. Using this energy, monks channel uncanny speed and strength into their unarmed strikes. As they gain experience, their martial training and their mastery of ki gives them more power over their bodies and the bodies of their foes.

They can run at super-human speed, even along water and vertical walls like it was as easy as farting. They have immense manoeuvrability, they can evade great hazards effortlessly, they can punch ghosts, they can catch arrows, they can punt ancient dragons across a room, they can shake off charms and fear like they were nothing, they can stunlock most enemies, and Open-Hand monks? They can one ONE PUNCH KILL A TARRASQUE.

So what in the white ghostly fluff is this nonsense you're sprouting?

I ain't no monkey-fan, much prefer my sorcies 'n' wizzies an' spellchuckers in general, really. But holy handbaskets in hell, the trashin' of the monk here ain't got jack to do with the class, bud. Either you're messin' up or the player just doesn't know how to play one.

An open hand monk can blink-speed to an enemy, stun them, trip them, kick 'em across a room and be back to their startin' position, safe 'n' sound,  all in under 6 seconds. If you can't make that seem "special" you're the problem.

Why isn't the cleric casting bane on the enemy and letting the monk go be the stunlock cheesemuffin they're designed to be? Bane them, the monk trips and stuns, free advantage all round - Monk MVP. 

Open Hand monks have "ultimate badassery cheesewalk" as a 3rd level feature. How poorly is the player playin this monk to not feel special when they're playing a class that can punch demigods?

The monk has a family. Use it. Her brother was kidnapped by slavers and taken to a land far away. Her grandmother died and her last wish was that her ashes be scattered in the four corners of the world. It lets the monk travel, and gives her and the party a reason to go to these places. Throw in a vision from a god or dream or what have you so that 1-2 other party members want to go to the same place. 

And I agree with cybermind maybe the player doesn’t understand how to play a monk. They can require a bit of subtlety and strategy, and can often be better setting up other characters. They could be in the back, stun-locking casters that others can’t get to. Or, while standing next to the pally and doing a d4 damage won’t seem useful, What is useful is knocking enemies prone next to the pally to give them advantage, so they crit more and can double those smite dice.

Maybe the player is nervous about using their ki points, so they’re just punching things instead of pulling off their fancy tricks. Sometimes, particularly with new players, they can be gun shy about limited resources. Could be they need to learn more about the resource management aspect of the game.

You don't have a Rogue.  It seems odd that a player who doesn't want their character to be magical in any way, and use their skills instead, would pick a Monk.  Who is finding the traps?  Who disarms them?  Is anyone scouting ahead inside the dungeons?  Do the characters just blunder into every single trap, ambush, or monster?

Monks can be sneaky.  They can learn to find and disarm traps.  They can scout.  They're speedy too.  They can come back and report in a hurry if needs be.  Of course, scouting becomes entirely irrelevant if anyone in the party happens to have a Familiar.

Monks have their place.  They tend to be all over the combat, booting heads, knocking people prone, stunning them and keeping them locked down so everyone else gets to boot head too.  Rogues are much better at being non-magical critters who use their skills for the benefit of the party.

You had me at, "Stunlock Cheesemuffin". 

To expand on "her grandmother died and requested that her ashes were scattered to the four corners of the earth":

Her grandmother was in fact a warlock, but a cursed one - the patronage is passed on to the youngest descendant when they die. The only way to break the curse is to scatter the remains of the previous cursed person to the four corners of the earth.

Until then, the monk is plagued with visions from the patron, and it has some element of influence over their decision making - whenever you give them something like "it seems like a good idea to _" or "the _ looks very interesting", these hints could be influenced by the curse.

Ultimately they will have to decide between accepting the gifts offered by the patron (and multiclassing into warlock), breaking the curse by traveling to the four corners of the earth, or living with the constant whispers and tempting visions offered by the patron.

I'm with Cyber on this one. Monks can do things that are not possible for creatures that have no access to magic. There's really very little difference between a monk and a sorcerer aside from how they express their innate power. Perhaps you could differentiate their arc from the warlock or sorcerer by making ki a new 'discovery' by the powers that be as a new and questionable type of magic that needs to be categorized.

Standard Tropes:

1. Apprentice becomes a godlike being.
2. Commoner becomes King 
3. Low level fights and beats massive Crime lord (may or may not replace him). 

It sounds like you do not want to do #1.  That leaves the others.

2: You can introduce a Kingdom ruled by Monkish people and have them ask for his help.   Repeated missions could have them respect him more and more, slowly offering items and jobs tied to political power.    Marry the princess and become King.

3:  Have him targetted by a crime lord.  Start it with a random thing, have the party attacked by five thieves.   Pick one that he gets the final blow on (cheat on rolls/hitpoints for the second to last one left alive till he happens to kill him if you have to).   Whoever he kills turns out to be the son of the big bad crime lord, instant feud.    Keep the Crime Lord after him specifically till he figures out how to kill the Crime Lord.   Up to you if he is offered the replacement position "You killed the Boss, so we are all obedient to you lest you kill us".

Sounds less like a magic problem and more like a lack-of-backstory problem. If you have nothing to throw at the monk because their entire backstory is "I left my happy life to see the world", then you either gotta talk to them about having some story hooks for you, or just make some up. 

To paraphrase Marisha Ray from Critical Role, "content people aren't the ones who feel the need to leave their happy lives to go kill dragons". You gotta figure out what makes them tick, so to speak

I read somewhere that the key to making characters feel special is to not look at skills like "Deflect arrows" and think, Ah, so I should never fire arrows at my monk because it would be useless.  Instead think, Damn, I need to fire arrows at my monk.  Give your monk walls to run across in the dungeon where the floor has caved out, or water to run across because they are the only one who can do it. 

This is a kingdom that is against magic users, introduce a special secret order of monks that works for the king who's job is to eliminate rogue magic users, especially high level ones.  The party must be gaining attention as they gain levels, make them like Ty Lee in ATLA, when they attack your magic players, if they fail say a DC14-16Con save, they can't use magic again until they pass the save, but your monk with the martial art training would be immune.  And if you want to go further, they are using a way of the Open hand Technique to disrupt the flow of Ki in the players to prevent them from casting magic, tell your player perhaps in a DM or with a Text message that your monk notices this (After one or two times of it happening) and might be able to reverse the effects by striking her companions (gently) and realigning their Ki.

Perhaps it's a bounty-hunter or someone who is after the party or their loot and they have an artifact that can create an Anti magic zone because they know the party is full of magic users but didn't realize there was a monk.  Think about Bad Guy motivation, "Who is against the party and what do the know about the party." Well, it sounds like this is a party full of magic users, and if I was up against them, I would be preparing to counter magic.  be that anti-magic areas or items that give bonuses to saving throws or resistance to magical / elemental damage.  However, they probably aren't prepared for the monk who (Despite being "Magic" as some people have not so nicely pointed out) isn't really affected at all by anti magic zones or magic resistance.

I would be worried that the Monk's ki-power abilities will be seen in your world as "magic from within", and thus the character will become feared/hunted for being "a person with inherent magical talents".

As a start, you could get the player to describe HOW they are focussing their KI in order to achieve anything that requires spending ki points. This will probably involve some physical movements that others could interpret as spell-casting.