I have a player who wants to wield a Warhammer in one hand while mounted, and is trying to keep the d10 for dice? Is there a feat or bonus that does it? If not would it be a problem balance wise if I let him do it?
Mounted combat lets you use versatile
There is no feat (that I know of) that lets you use the increased damage you get from wielding a versatile weapon with both hands while you wield it with one hand.
However, this should not be an issue for your player: the rules for controlling a mount (p. 198 PHB) do not say that it needs one of your hands, so it does not. (Just think of jousting knights with lance and shield, for a narrative justification). So your player can wield the weapon with both hands while mounted, and get the higher damage.
There also is a Mounted Combatant feat. While it does not do what you ask for, it provides various other bonuses for a mounted combatant like advantage on melee attacks against unmounted creatures smaller than your mount.
From a balance perspective, a feat that would allow this would be weak. Effectively, upgrading from d8 to d10 is worth slightly more than 1 point of expected damage (the little more is from doubling the dice on critical hits). Just taking an ASI to increase your strength likewise will increase your damage by 1, but also will increase your to hit by +1, which is worth about as much damage again, do this for all melee and thrown weapons, and increase strength ability checks like grappling, climbing, swimming, forcing open doors by +1, improve (rare) strength saving throws by 1, and improve your carrying capacity. So such a feat would be weaker than half of what a feat should be. I at least would make it a half-feat and also let it increase Strength by 1, and that would not be unbalanced.
Just letting them do it while mounted would also probably not break anything in a major way. It would be weaker than letting them start with a magical +1 weapon (which again would add to both attack and damage, and has the added bonus to count as a magical attack), something I've seen done that has not broken those games.
But I feel it is a pretty strong thing to grant for nothing, especially early on where your expected damage is likely less than 10 points, so it increases damage by more than 10%, for free, and later on you can combine it with all the other ways in the rules to improve damage. Or it allows them to use s shield for +2 better AC, which is huge, while dealing 1d10. There is no melee weapon in the game that does this, so I'd be careful.
As @Slaves_of_the_Coast points out, you can however use a lance with one hand as a mounted combatant, which deals 1d12 damage. Attacks on opponents within 5 feet with it have disadvantage, but that can often be avoided if not fighting in close quarters, so it might be an interesting, rules-based option for your player, too.