Chaosmancer
Legend
I'm ambivalent about hit dice being used as a resource to power more things.
I agree with @Kobold Avenger and @codo that it would fit the fiction of expending life force or pushing beyond normal limits to do something, which is cool, but I don't think that kind of feature would work very well with the current implementation of hit dice and resting.
In my mind, the biggest issue with hit dice is the way they fit into the resting rules. They have no value during long rests, since long rests fully recover HP, and can only be spent on short rests, which most classes don't much care about and a lot of groups don't take very often. As a result, they're a mechanic that, depending on party composition and playstyle, is pretty forgettable, "vestigial" as others have said. That means that new functionality for hit dice is gonna be inherently kludgy--and will also vary a lot in usefulness by party composition and playstyle.
Ideally, I think, PCs should be able to spend hit dice to recover HP whenever they're out of combat to keep their HP topped off (which necessitates you to think of HP as luck points or not getting hit points, but no need to rehash that debate). And, since that would functionally eliminate short rests for several classes, it would be best if paired with short rest classes transitioning to proficiency times/day, at will, and/or encounter features (a direction WotC sometimes seems to be moving toward anyway).
I solved the "fitting into resting" issue by removing restoring to full hp by long resting and replacing it with rolling hit dice. In theory, this creates a big chance for me to grind people down and drain resources over time, and make Hit Dice THE most valuable resource.
In practice, after a dungeon people often have 2 to 3 days of resting, so they effectively end up at full anyways. But I think of that as a strength, as if you have a few days of downtime, then you should be back up to full. It is only battle day in and day out that saps you towards exhaustion.