My solution was to have no daily limits to adventuring, so this is how I fixed HP. (I've got other ideas for fixing daily powers to fit my style.)
HP represents your stamina, and when your stamina runs out, you are vulnerable and can't defend yourself. But you're heroes, so if you don't die right away, you'll feel better quickly.
You have HP as normal, plus you have unlimited healing surges. Your healing surge value is the same as in normal 4e.
Out of combat, you automatically heal to full HP after 5 minutes of resting.
In combat, there are four ways to heal.
A. Getting your second wind. As a standard action, you can make an Endurance check (DC 10). If you succeed, you trigger a healing surge and your defenses get a +2 bonus for one round.
B. Mundane healing. As a standard action, an ally who is adjacent to you can make a Heal check (DC 10) to trigger a healing surge.
C. Bolstering words. As a standard action, someone who you can hear and understand can make a Diplomacy check (DC 10) to trigger a healing surge.
Each time anyone uses one of these options, the DC for further attempts of any sort increases by 10. So if you get healed once, it's DC 10. Then if you take a second wind, it's DC 20. Then if a buddy tells you to keep fighting, it's DC 30.
Finally, D. Healing powers. If someone has a power to let you spend a healing surge, that works as normal, without requiring any sort of check. Healing surges granted by these don't increase the DCs of the other options.
I also wanted HP to work so that when you reach 0 HP, you don’t fall unconscious. Instead, you are stunned, but each turn you can make an Endurance check (DC 15 [perhaps 15 +5 per failed death save]) to just be dazed for that round. You still make death saves, and when your HP falls to negative bloodied, you still die.
I'm considering having some sort of very simple "wound system," which would be that, if you fall below 0 HP, you start on a wound track, which would work like a disease track. Each time you fail a death saving throw, it would get worse. Each extended rest would let you make an Endurance check (DC to be determined) to improve one step.