I've been preparing uses for Hit Dice other than Healing and the idea of getting rid of exhaustion is nice.

I prefer to make it so expending a Hit Die means rolling it, pretty much no matter what it's being rolled for. Based on that, I'd probably set a DC to removing the level of Exhaustion -- I'd math it over before playtesting it, but probably in the 5-10 range -- and if the value of the Hit Dice rolled (probably adding your Constitution Modifier, in which case the DC is higher) exceeds the Exhaustion DC they lose the level of Exhaustion. This has the nice little benefit of martial classes being a bit more likely to heal their Exhaustion with less resource expenditure, which appeals to me given (a) they could easily be dying more often, and (b) they tend to be the classes that would've trained themselves physically a bit more (not that Exhaustion is limited to that!).

Depending on the power level desired and with the caveat that they have to expend that class's Hit Dice (in case of a multiclass), you could do something like:
- Paladins being about to use a number of Hit Dice up to their Proficiency Modifier for Divine Smites when not using Spell Slots (powerful, but remember they can only regain up to 1/2 on a Long Rest)
- Life Clerics being able to supplement Healing (others included) with their Hit Dice
- Abjuration Wizards being able to expend Hit Dice (maybe capped at their Proficiency Modifier per use, potentially with a bonus) to reduce Damage they take or to Heal their Abjuration Ward
- Necromancers can Heal their Controlled/Summoned Undead (or give Temp HP?) using their Hit Dice (helps make up for it being so hard to Heal Undead)
- Transmutation Wizards and Druids can use their Hit Dice to Heal themselves while Transformed/Wild Shaped
- Conjuration Wizards can use their Hit Dice to Heal their Summoned Creatures
- Wild Shaped Druids can expend Hit Dice to change their form, with their HP being the lowest of [current HP + Hit Dice total] and [new form's Max HP + Hit Dice total]
- Bards can use Hit Dice instead of Bardic Inspiration Dice for some of their abilities (which and whether would depend on the Subclass)
- Wild Magic Sorcerers can expend Hit Dice to create Wild Magic regions around them (...if you like chaos, confusion and probable low-level death :P)
- Fighters can expend one (or more) Hit Dice when using Action Surge to get a second Bonus Action. If they make a Weapon Attack with that extra Bonus Action, the Hit Dice total is added to the Damage
- Monks can expend Hit Dice to regain Ki (probably using an Action, to make it less appealing, and perhaps half of their total rounded up)
- Rogues can expend a Hit Dice when they fail a Dexterity Saving Throw and, if their Saving Throw total + the Hit Die total allows them to pass, they pass (better) or only take half damage (kind of pointless given Uncanny Dodge, unless this ability doesn't require a Reaction)
- Sorcerer regains Sorcery Points using (probably using an Action and half the roll, like for the Monk [the different Hit Dice sizes helps adjust for Monks getting Ki more easily than Sorcerers get Sorcery Points])