On the one hand, I cut my teeth on D&D, and I think that hit points, along with the original six character stats and character classes (and i'm sure there are others) are essential elements of the "D&D" game.
On the other hand, I have accepted, by never been terribly satisfied with, the mish-mash definition of hit points, being a mixture of physical punishment, endurance, luck, sixth sense, and so on.
I ocassionally play Lord Of The Rings: Strategy Battle Game. While this is no RPG, it is an interesting "rules-light" tabletop wargame. Each of your units is an individual warrior. Typically, a warrior has only one "wound": you hit him, he's dead. All of the weapons do one damage, regardless of weapon type. A hand weapon is a hand weapon, whether it is a sword, a mace, or an axe.
Where LOTR:SBG gets interesting is with the "heroes" of the game. While rank and file warriors have one wound, heroes also have "fate" points. A hero with one or more fate points can use a fate point to avoid taking a wound, as a result of a successful attack. While a hero can only have a maximum of three wounds, there is no maximum number of fate points a hero can have.
I find this approach attractive, as it seems more "true" to the idea of heroes, by recognizing that a hero can anticipate a blow and thus avoid it at the last second (by using a fate point) or that the gods themselves protect the hero, as it is not his "fate" to fall in this battle. Whether you could apply this approach to D&D is another matter, but it would be interesting to try.