I like the concept, but I think you may need to come up with a list of abilites that can be reused. That way the players don't find a 'gotcha' that makes them overpowered. Add things to the list by the players request after you have a chance to think about it.
In novels, you always see the hero finding a way to push past human endurance to come up with the strength needed to win in a situation where they are in over their heads. I think that's what inspiration is supposed to be as well.
What I would worry about is having all the players suddenly using their daily abilities over and over when they know they get to the big bad boss fight. There are also some classes (like a rogue) that would not benefit much from this.
I may allow something like this at my table. Here's what I'm thinking:
Take a level of exhaustion to:
- Regain a spell slot (Spell level 3-4 is two levels of exhaustion, spell level 5 is three levels. Not avialable past 5)
- use a spell slot to cast a spell not memorized
- re-use an ability that refreshes on a long/short rest (subject to DM approval)
- re-roll a saving throw (including death saves)
I'm not sure how I would handle superiority dice, ki, or rage. Should the player get all uses back, just one, or not have these be on the list as rechargeable by taking exhaustion?