We all know about rolling up all the hit dice every time you level up, which is a fun lottery thing you can do. When you level up, roll all your hit dice (including pluses if that's how your progression works). If that number is higher than your current total, use the new number. Otherwise, just add one hit point to your old number.
I had also read somewhere, I think on DF, that some people would roll up hit points every morning of the campaign. That sounds fun and risky. Not sure how it would work with hit point damage though. Maybe we would need a different philosophy about what a hit point is.
It just came to me that you could roll hit points anew every day. BUT instead of in the morning, you would roll them when you first took damage that day. That way you really wouldn't know exactly how healthy you are until it was put to the test!
After all: creatures you never fight don't have hit points. They only have the stats that they need (usually none or just a couple.) They only get hit points if you need to fight them. Could we confer this property of hit points to the PCs and their allies as well?