In prior versions I think nunchucks fulfilled this category, but nothing exists in 5e? Are there maybe batons or some other finesse weapon that deals bludgeon damage in an expansion I dunno about? Oh. Or by chance are there variant rules that make fists finesse weapons that deal bludgeon damage?

Nothing official, which is something that I've found annoying.  You can use dex with monk weapons, but that's explicitly not the same as finesse, which means you can't make sneak attacks with them.

Making an actual finesse weapon that deals bludgeoning damage requires house rules/homebrew.

Aww I was afraid of that. It is the same with slashing. I cannot find a finesse simple slashing weapon.

Yeah, the whip and scimitar are the only two finesse weapons that don't deal piercing damage.  And the double-scimitar if you've got the correct feat for it.

It doesn’t break anything to change a dagger’s damage to bludgeoning and call it a blackjack or something like that, if your DM agrees.

Unless you’re dealing with actual resistance to piercing, the ability to incapacitate / knock out melee targets that drop to 0 HP can be described as a pommel strike or similar. Is there a problem you’re trying to solve by dealing bludgeoning damage?

I am a martial type and I want to be able to diversify my damage types to overcome resistances. I know that for martials it isn’t as important as it is for wizards maybe but I am role-playing a tactician type character. I took a Monk level for the unarmed/unarmored abilities and I want to stack it with my sneak attacks across each damage types.

I am using currently a mace for bludgeon and a handaxe for slashing. Both are Monk weapons but neither are finesse, then I have a shortsword for my piercing damage type and it is both a monk weapon and finesse so I can sneak and add the monk unarmored follow up. I am 100% positive that will make the shortsword my primary weapon. All are 1d6 so it is nice to have uniformity.

There really should be batons and kamas for finesse slashing weapons. They were a thing in 3.5e. I wonder why they removed them?

What monsters are you facing because aside from skeletons or certain oozes I rarely see any need to distinguish between piercing/slashing/bludgeoning. Most creatures with resistance to one have resistance to the others as well. While there are some monsters with specific resistances/vulnerabilities they are very few and far between, so it's weird you feel the need to "overcome resistances". 

Yeah, wizards have to prepare spells with diverse damage types, because quite a lot of creatures are resistant, immune, or vulnerable to one particular magical damage type. But most creatures that are resistant to one of the three physical damage types are resistant to all non-magical physical damage.

That being said, it really doesn't break the game to take some homebrew finesse weapon with piercing or slashing, and make a homebrew finesse weapon with the same properties and damage dice. Whip + bludgeoning = meteor hammer? Sure, why not? Like it's not going to break the game if you sneak attack an ooze without tripping its Split ability, which you can do with a rapier anyway.

The Weapons table shows the most common weapons used in the fantasy gaming worlds, DM and player are more than free to reflavor or houserule more weapons indeed. I had a boken in one of my campaign that was a simple finesse club sword made of wood.

I know it is silly because like you point out almost everything is resistant to all three types of damage if they are resistant to one type, but I want to at least role-play having the right tool for the job in battle so I still want to best sort out my damage types. I don't know maybe it is stupid, but I kind of like the idea.

I know I was the last to post but I figured I'd give it an update since playing the character.

It was awesome. I had a handaxe, mace, shortsword, and spear. The handaxe and spear were my ranged options; I do want to get a heavy crossbow, and the mace allowed me to keep the final battle of the session non-lethal. Instead of killing the bandits we apprehended everyone and then since I am a noble one of my party members pressed them into my service as mercenaries.

I'd still like to improve the build, but having so many weapons was really fun.

In 3.5 a Sap was  a 1d4 damage light bludgeoning weapon, basically a blackjack.

Quote from KingWalla46 >>

In 3.5 a Sap was  a 1d4 damage light bludgeoning weapon, basically a blackjack.

 Under 3.5 rules, you could use Finesse with any light weapon, like light maces and light hammers.  The big difference with the sap was that it dealt nonlethal damage by default.

Quote from 6thLyranGuard >>

Quote from KingWalla46 >>

In 3.5 a Sap was  a 1d4 damage light bludgeoning weapon, basically a blackjack.

 Under 3.5 rules, you could use Finesse with any light weapon, like light maces and light hammers.  The big difference with the sap was that it dealt nonlethal damage by default.

 This is true.

I know it's not a melee weapon, but you can technically use Sneak Attack with a sling for the bludgeoning damage there.

Quote from SagaTympana >>

It doesn’t break anything to change a dagger’s damage to bludgeoning and call it a blackjack or something like that, if your DM agrees.

 
Or to take a club, give it finesse, and call it a nunchuck. That’s what I do at any rate.

The game works fine if rogues can sneak attack with anything they're proficient with, including their legacy longsword proficiency.  There's no critical game math that forces rogues to stick strictly to light blades.  Wanna sneak attack with a mace, or a quarterstaff, or a hatchet?  Go for it.  The math will still work perfectly fine, and you may see some really neat unusual rogue builds come from it. 

Quote from Yurei1453 >>

The game works fine if rogues can sneak attack with anything they're proficient with, including their legacy longsword proficiency.  There's no critical game math that forces rogues to stick strictly to light blades.  Wanna sneak attack with a mace, or a quarterstaff, or a hatchet?  Go for it.  The math will still work perfectly fine, and you may see some really neat unusual rogue builds come from it. 

 Monks need light weapons. The build is Monk/Rogue.

Monks don’t need light weapons or else they couldn’t use a quarterstaff.