When playing RAW, the only simple melee weapon with the finesse property is the dagger (PHB page 149). Most other thrown weapons of this type (hand axe, javelin, and spear) all do 1d6 damage with a secondary property thrown in — hand axe is light, javelin has extended range, and spear is versatile. This makes the light hammer — 1d4, light, thrown — inferior to these other thrown weapons in damage, and short of the dagger in its lack of finesse. As written, I can't see why anyone would choose it. Is this broken?
People elsewhere have discussed increasing light hammer damage to 1d6 as a possible homebrew fix. The typical objection I've seen is its potential for use against monsters vulnerable to bludgeoning attacks (e.g. skeletons). I am willing to concede this argument, though to me, the RAW light hammer still feels imbalanced vs. hand axe — most of the time, damage type just doesn't matter.
The historically-appropriate change might be some kind of attack/damage bonus when hammers are used against solid armor types (breastplate, half plate, or full plate), but for this change to be meaningful you'd have to do the same for mace, war hammer, and maul — and none of those weapons need it.
My thought: could we instead assign it an additional weapon property to compensate for the lesser damage (perhaps as a race feature, if not for everyone)?
For characters of dwarven (and perhaps forest gnome?) ancestry, for example, it seems more race-appropriate to use light hammers in place of daggers, and if they were a finesse weapon, dexterity-based characters might do so. Is there anything specific to bludgeoning damage that would make this a bad idea? Dwarven rogues would thank you if it were done.
Best Answer
Not all weapons are equally powerful and they don't need to be
The light hammer is not the only weapon that you would never use from a perspective of power.
The sickle is equally bad as the light hammer. Similarly, there is really no reason to specifically use a trident. It is the same as a spear but needs martial weapons proficiency.
When you start to consider that every weapon should have their advantages, you have even more work.
Then again, I don't see why it is absolutely necessary to only have weapons that have a use for the optimizer.
Finesse seems very non-thematic on a hammer
While adding finesse to the light hammer won't have many mechanical consequences, it means that Dex based characters now have a finesse bludgeoning weapon against skeletons. It is true that this probably won't come up that often and you said you don't find this important but it is a fact. It won't break your game.
On the other hand, as stated above, I don't think there is a problem to be fixed. And adding finesse to a hammer seems very non-thematic.