Unfortunately I do not have an idea for using HD for fueling effects, but perhaps this re envisioning of HD might give ideas.

Stepping away from HD at first, imagine giving every character the maximum hit points.

Divide these into two HP pools. The first gets treated like HP regularly, the second only gets damaged on a critical hit where the regular damage roll (So a critical dagger strike does 1d4+Dex to the 2nd pool or 2d4+Dex to the 1st pool with bleed over into the 2nd when the 1st is depleted)

Now the 2nd pool of points can be used to spend to get HD to roll and recover and charge the 1st pool of HP, or spent on other things (perhaps convert to spell points). I like the point idea, because you can assign point values to various dice (2 = d4, 3 = d6,..6 = d12) and then allow recovery time based on dice rolled (and probably other mechanics required). It may be too much work to make useful, but I like the idea that you have to balance keeping vitality up and your recovery resources. If you recover half of 2nd pool back on a long rest, you then take time to recover from a serious fight, and could be set back by a serious blow.

Basically a Vitality and Wounds system that makes HD idea of 5e actively considered throughout the game.