I'll be starting a group with a Paladin, a Monk, and a Druid. What class would you suggest the fourth player (me) play as to best fill out the party during Mines of Phandelver?

Something skilly or something arcane. Rogue, wizard, sorcerer, warlock, bard or alchemist, oops, I mean artificer, could all work. What do you feel like playing?

Would also say skilly or arcane, and definetely ranged.

Quote from Lisztkrieg >>

I'll be starting a group with a Paladin, a Monk, and a Druid. What class would you suggest the fourth player (me) play as to best fill out the party during Mines of Phandelver?

Artificer, Warlock, or Wizard.

You've got a pretty well-rounded crew... one dedicated caster, two diverse martials, and two characters with access to healing abilities.

I think you could use a Wizard more than anything else. I like Rogues, but not to get too metagamey, but there's not too very much "Rogue Stuff" in Lost Mines of Phandelver. There's some secret passageways, a few traps... the occasional locked door. Mostly stuff that a wizard with the Knock spell can take care of. But you get a lot of utility as a Wizard plus some big damage dealing spells. Evocation Wizard is probably the most popular choice for newer players... the ability to just exclude your allies from AOE spells makes them a lot easier to use, but all the Wizard subclasses will perform well. My personal favorite is probably Divination Wizard, since it gives you the opportunity for some pretty epic reversals of fortune.

Warlock,  Bard or Wizard. are the prime choices.  Warlock does better at lower levels, but suffers in mid levels, but suffers later.  It is not really a primary caster, but more a half-caster - not enough slots to be full.

Bard is harder to play, but will take of the party missing skills as well as spells.

Wizard is the easiest to play and will rule the mid level on.

I will also say that you have a pretty balanced three-person party with all your friends already, so you don't have to worry as much about fulfilling a specific role. So you can take advantage of this to play just about any class... if you've got a weird build you're drawn to, there's nothing holding you back.

5e is pretty foregiving with group composition especialy if the DM bears the group in mind (eg if noone can heal I would expect the DM to put more healing potions in the loot). The most important thing is to play a character you will enjoy playing with (without upsetting the rest of the party). Having said that it is good to have complementary skills if everyone has good social skills you can feel you are missing out when someone else is acting as the face.

If the Druid is going to be circle of the moon you have 3 PCs that will mostly be melee (at least from level 2 until you finish lost mines) so I agree something ranged is a good idea. It can also be good to have someone wth high intelligence who is likely to know things about arcana, religion, nature and the likes. That would make wizard and a ranged artificer (artificer builds are very diverse) though I would say neither are the easiest classes to play for a new player.  If you want to go with something simpler you could go rogue, Ranged yogues only really need a high dexterity (though they shouldn't completely dump con) so you could still have a high inteligence which along with a lot of skill proficiencies should make you pretty good for ability checks. Arcane trickster is the obvious subclass with high inteligence but you can go with any subclass.

Like others, I'd suggest looking at the Bard or Wizard. Lore Bard or Eloquence Bard would probably fit in well with this party, but an Enchantment or Evocation Wizard could also fill gaps I think the party might have (don't know for sure without seeing everyone else's builds). I assume the Paladin will be the Party's face, but because they will need to build up Strength and Constitution they may not be as capable in the Face role as an Enchantment Wizard or any Bard. The Wizard choice also gives good ranged damage and utility options for the party (they will love you for casting Invisibility or Fly). The Bard options will give you lots of good skills and allow you to be the party face, as well as some good buffing for your teammates and debuffing for enemies.

I would say something like rouge or artificer (just my preferences) or a spellcaster like sorcerer or wizard

Arcane Trickster Rogue would fill out a bunch of "missing" bits, getting a bunch of proficiencies for the skill monkey, a little arcane ability (could dip Wizard for Ritual casting at some point) and another solid melee skirmisher for the group.  All other options work well, too, as the existing trio is pretty rounded and capable already.  Bard, Wizard, Sorcerer all work and fit and will contribute fine.  Pick whatever really tickles your fancy and I'm sure you'll end up being a welcome addition and contribute throughout.

If you have the stats, then rogue 1 wizard rest of levels. If the GM allows Strixhaven then take witherbloom as your background and you can fill the rogue roll, wizard role and witherbloom gives you healing.

Have Fun

When it comes down to it, I would say choose whatever you think would be most fun. But if I made a choice based off what class you already have... I would pick a ranger. They have a range of perks such as favoured prey and the terrain thingie, as well as they can use spells and they make up for your current party not having a ranged fighter.

As well as all of that ^^^, At later levels you can choose the subclasses. I find that using the "Hunter" subclass is beneficial as it allows for attacking more than once per turn despite normal actions per turn, or can allow you to deal more damage per turn. Also, since you have ranger class, you can add your dexterity modifier to your AC I'm pretty sure and since you would naturally try for higher DEX as a ranger, using a finesse weapon could also benefit as it would allow for more damage using a similar effect to the previous one. (At least I'm pretty sure this is all how it works, I don't really read the outline that well, I just skim it for good Buffs and such to make my character more OP.)

This is just my personal opinions and to be completely honest I don't have much knowledge when it come's to class distribution and the such. Me and my friends usually just make whatever characters we like and that we will have fun roleplaying and using. Feel free to disregard all of my input completely if it seems useless to you or anything like that.