I love how simple hit points are as a mechanic. I hate how abstract they can be, especially as levels add up and totals rise. I love how easy it is to go from "6 damage? Run for your lives!" to "6 damage? Whatevs, I got this." I also hate it.
I love how, at low levels, players have good reason to avoid battles because one or two wounds could easily prove fatal. I hate seeing players burn through PCs like extra lives in a circa 1985 arcade game. (But I kinda love it, too.)
I love how, at high levels, players can press on through more dangers and get themselves deeper into the dungeon before they have to turn back and hope they didn't push their luck too far. I hate how the mechanic pushes toward either a slog of multiple melees in which the PCs are gradually whittled or a tac-nuke monster that barely cares that the PCs have leveled up.
I hate how environmental hazards, like fire, falling, and traps, wind up being less dangerous at higher levels just like melee. I hate how healing spells are less effective on high-level characters than on low-level ones.
I've tried to get around a few of these issues in my campaign. For example, I base most healing on hit dice - a Cure spell restores hit points equal to 1d6 + the recipient's hit dice (e.g. if you have 3 HD, it's 1d6+3). Likewise, a hazard that doesn't care how good you are at taking a hit in combat will roll your hit dice in damage (instead of "Save or die," you might call it "Save or possibly die, depending on the luck of the dice, and if you're already wounded the odds are against you, but hey you might get lucky").
I like the game being pretty gritty, but I like the odds in favor of survival - if the party survives by the skin of its collective teeth, I feel like everybody gets the maximum excitement. I respect the mathematical architecture of 4e - or Type IV, or whatever the kids call it nowadays - for providing that experience pretty reliably, but that math is pretty fragile, and a simple thing like changing the number or composition of the party can have dire consequences (this is what killed my attempt at a 4e campaign in its infancy, and eventually led me to the OSR).
I like to ramble, but I hate the fact that I'm rambling (and I like to end paragraphs with pointless parentheticals, for some reason).
So here's an idea the combine gritty, survive-by-the-skin-of-your-teeth action with that-was-exciting-now-let's-keep-going adventure. The idea is that PCs begin with a Hit Die full of hit points, as usual (and you can use your favorite Hit Die rules - max at first level, reroll 1s, variable or standard, whatever). When the PC would normally gain a new Hit Die, he adds +1 to his hit point maximum and gains a "reserve" point. A reserve point can be spent to recover a Hit Die's worth of hit points. You can't go above your maximum, as with any hit point recovery. Spending a reserve point this way should take a full turn of rest.
The idea is that every battle remains gritty and stark, although the PCs will gain an edge in toughness over time; but the more remarkable and heroic development is the reserves of strength that enable them to carry on even when hurt and exhausted. I envision a scenario in which the party finishes a battle and spends the rest of the round catching their breath and binding their wounds, and the players roll their reserve dice to discover who was simply running out of stamina and who has a serious injury.
I wouldn't implement this system for monsters. Monster Hit Dice are more related to the size and toughness of the monster than its melee skill, and there's no need for every PC mechanic to be paralleled in monster mechanics (that way lies the excess of 3.x). I'd probably give Reserves equal to HD to any monster that's a named individual (e.g. Brok the orc warlord, Anton the evil high priest, Skaldar the legendary ogre, and Azraphael the dragon). Everybody else is an extra, and runs by the book.
I'd love to test this system out, but I'm pretty sure my regular players would rebel if I imposed it on their long-standing characters. Maybe I can test it in some side project, like a scifi one-shot or something.