Some of the comments in this post gave me a stupid idea. Using hit points as magic points seems like a neat idea. It plays into the idea of hit points as luck, divine favor, and so on. It gives magic using types a resource to manage that is non-trivial to balance against other considerations. It's just that, as others noted, creating a whole new magic system is a pain in the ass.
Fortunately, there already exists a magic system written for a version of D&D that uses hit points as the magic points of the system. In the D20 version of Star Wars, the Jedi and other Force users use hit points to power their abilities. Now, it would need some conversion to go from D20's skills and feats (which is what the Force system in D20 Star Wars is built around) to regular D&D terms, but at least the conversion is possible. Instead of spell books and spell slots, these sorcerers (we might as well call them that) would learn things at each level that are analogues to the skills and feats of D20 Star Wars.
Not something I'm going to do right now, but just thinking about it. Another magic system that I'd like to write up one of these days is also inspired by those comments, but it would be even easier since The Palladium RPG did a version of it, and that's the magic user who gains powers from pacts with demons and other spirits.