Hello 👋, I'm a relatively new dm. One of my players (a warlock) has been using an axe (great axe i think) for a good while now and is really attached to it so I thought, instead of giving him a new magic item it would be a good idea to have his patron or someone similar magically bless his axe or increase it's power for completing a quest.

How would I go about doing something like this? All help would be greatly appreciated and thank you very much in advance to all those who respond

I think you're already on the right track. Give the player some sort of quest from their patron (that takes a decent amount of effort to do) and then give them some sort of thematic upgrade to the item.

You can homebrew a weapon for them basing it off of a weapon, +1 and add some form of thematic feature such as:

• An additional cantrip that fits thematically that they can cast through the weapon
• A spell that they can cast once per long rest for free through the weapon
• A selection of 2 or 3 spells and charge costs, and a some amount of charges
• Change the damage type of the weapon
• Bonus damage to a certain type of monster

What's their subclass/who is their patron? What's their pact boon? If you give a bit of info, I can give some more specific examples :)

Thanks, that's great help. The character's a dwarven hexblade warlock. His patron is a extremely powerful red (but good aligned) dragon who's lair is a volcano beside a frozen dwarven mining city and has many kobold servents.

Give some flavor to the end of the quest. Something like:

As you [fill in some action that completes the quest], you know that your actions have pleased [the patron's name]. A lash of flames briefly swirls down your arm and around the hilt of your greataxe. You feel some warmth within your axe, and see that the handle becomes coated in red draconic scales, and the double-sided blade takes the form of dragon's wings. The weapon now occasionally pulses with a lavalike glow.

Depending on the level of the party, you could do any of the following:

• Homebrew a greataxe based on a flame tongue (low to mid level party)
• Homebrew a greataxe based on a staff of fire (mid to high level party)
• The weapon is now +1 and allows the player to cast fire bolt from the weapon (low level party)
• The weapon is now some form of +X weapon, and the player can cast a level-appropriate spell from it such as: 
  • +1 and burning hands once per long rest (low level party)
  • +2 and wall of fire once per long rest (mid level party) 
  • +3 and draconic transformation once per long rest (high level party)

Those spells listed are just examples, lots of different options based on what you think would fit for the general power level of your party.

Some other spell suggestions:

2nd level: Aganazzar's scorcher, dragon's breath, scorching ray

3rd level: Ashardalon's stride, fly (flavored as dragon wings)

4th level: fire shield

5th level: immolation

One final suggestion, If you have Fizban's Treasury of Dragons, you could give them a dragon's wrath weapon. That would allow it to keep levelling at certain plot points in the campaign.

All great stuff from Javapiston14 above. I'd also suggest some passive features too - perhaps resistance to fire and cold damage while attuned to the weapon, as the blade heats the wielder against the cold and is a firebrand itself. 

Use the Ruby of the War Mage as inspiration. It’s a gem that sticks to a weapon and enhances it. You can homebrew something similar. 

What level is the character? You may want to start with just making it +1, and save the add one for later. You don’t want to give out super powerful weapons at low levels. 

You might check out the wildemount book. They have vestiges of divergence in there, which are weapons that power up as character’s achieve certain goals. It might have some ideas you can steal. 

Quote from javapiston14 >>

One final suggestion, If you have Fizban's Treasury of Dragons, you could give them a dragon's wrath weapon. That would allow it to keep levelling at certain plot points in the campaign.

 This. It's simple starting out, there's a built-in mechanic for improving it, and it's far more interesting than a regular +1 sword.

I'm a little confused actually. Is the character level 3 yet if they have been playing a while? Are they a blade pact hexblade?

A blade pact warlock can summon their pact weapon which can take the form of any weapon. With the improved pact weapon invocation the weapon becomes a +1 if it doesn't have that benefit or better already and if using a generic pact weapon it can also be a bow or crossbow.

So at level 3, a hexblade blade pact warlock can summon up their own personal +1 Greataxe at will.

Creating a magic weapon for a hexblade is often thematic given the standard patron description (though in this case the patron is a good aligned red dragon so I would guess you are just using the hexblade mechanics not the associated lore which makes perfect sense in this case).

In this case, I think I might combine some of the suggestions above though you would have to see what is balanced for your campaign.

You could consider modifying the Improved Pact Weapon invocation in this case so that it can summon a Dragon's Wrath Weapon. Given that a Hexblade already has a +1 weapon from the invocation, I'd probably start with the Stirring version that would add a d6 of fire damage to the weapon based on the red dragon patron. As the character levels up or completes some quest for their patron you could improve the invocation to the next level of Dragon's Wrath Weapon.

This way, a blade pact warlock could keep using their pact weapon and it would remain relevant. Most warlocks end up making a magic weapon they find their pact weapon since it will be better than anything they can get with the invocation. This might be a cool way to bypass that and allow the character to have a very flexible Dragon's Wrath Weapon that can assume the form they want when they need it.

Though ... this does depend on the character being a blade pact warlock :)