Everyone has a theory about what a "hit point" is. Here's mine: bupkiss. Hit points don't represent anything. They don't make sense, and you can't make sense of them because they don't model anything in reality. This is a problem when trying to make rulings at the table, as you can't apply common sense. It also makes it difficult for other mechanics to work with them.
Any theory that attributes "skill at avoiding damage" to increasing hit points is nonsense. If that were true, then an unawares attack (like a backstab) should deal the base damage times the character's hit dice to account for the fact that the character cannot use their skill at avoiding damage. Also, heal spells should scale, damage healed per day should scale, all kinds of things should happen that don't.
These things are problematic because (much like with AC) you're mixing mechanics. Any increase in ability to avoid damage should come in the characters defense attribute, not their hit points.
And any theory that says a leveled character can actually take ten times more physical damage than a normal person doesn't know much about biology. It's just not reasonable. (Yes, yes, Rasputin. You can always find outliers if you look hard enough...).
Another problem I have with hit points (which I touched on earlier) is the loss of mystery. I'd like a little uncertainty about where a character is at, but I don't like the idea of having the GM track hit points secretly and only tell the players generally where their character is at.
The issue of stats meaning more than one thing is solved by moving hard-to-hit to Defense, hard-to-hurt to Armour, and your actual ability to take a lickin' and keep on tickin' to Fortitude. Fortitude is a derived stat based on your natural strength, health, and mental discipline.
Each time you're hit and damaged, you'll accrue Wounds, and make a check against your Fortitude penalized by the number of Wounds you have. If you fail the check, you're overcome by your wounds, and are hors de combat. This could mean passed out, you've fallen and you can't get up, or the pain is too much to bear any longer. In any case, you are not combat effective, and require medical attention. You'll accrue another Wound each minute until you receive assistance.
You'll keep track of Wounds much like you keep track of dwindling Hit Points. If your Wounds are greater than your Fortitude, you're dead. Time, medical assistance, and healing magic can reduce your number of Wounds.
This concept can be extended to fatigue, as well, although I think this should be an optional bolt-on. Under certain circumstances, you can accrue Fatigue - running, fighting in plate and mail, carrying ten stone of loot, and so on... Each point of Fatigue would penalize your Fortitude checks the same way Wounds do, and if your Fatigue + Wounds is greater than your Endurance, you've "hit the wall", and are more or less incapacitated from over-exertion. Rest, food, and water are prescribed. Or magic.
This mechanic immediately opens up an interesting idea for a feat or class feature - you can ignore your first failed Fortitude check, making you able to stay in the fight longer, but also more likely to suddenly die in combat as you push yourself further than normally possible.
This system eliminates the problems with the traditional hit point system, provides a workable mechanic, and allows for feats or class features to "hook in" in interesting ways.