Hit Points are used as a way of measuring stress in role-playing games. Games attribute different meanings to a character's HP, and some games have multiple tracks that measure different types of stress/damage. Normally there are special events that happen when the stress levels reach certain points; in D&D a character becomes "bloodied" at the halfway value, and most games have something that amounts to incapacitation upon reaching 0 points (or maximum points, if it is tracked that way).
Lord of the Rings Online uses Morale for hit points. The idea is that the entire motivation for a character to keep at it, fighting the enemies of Middle-Earth, is summed by the combination of stamina and will. It fits well, thematically. LoTR has a lot of plot surrounding morale and will, so they chose to abstract their health track in that way. It is particularly well fit when you consider the primary "healer" class in the game, Minstrels, who "heal" by inspiring their fellows.
Another interesting way to deal with HP (not to be confused with the Wizardling of the same initials) is by making it very limiting and somewhat more realistic, such as with FATE or the World Of Darkness systems. Those stress tracks have a little over a half-dozen "levels", and as you receive damage there are active penalties, which take the form of negative modifiers on dice rolls. So, fighting is brutal, and it is easy to incapacitate a foe, while also requiring everyone to be relatively conservative in the risks they will take. While there are "tank" builds in those game, they are the exception.
Dungeons and Dragons takes a middle ground with the amount of points it grants, while really going nuts with the abstraction. In LOTRO you will have many thousands of HP by the end game. In FATE you will have the same amount the entire game, but will have more choices in protecting them. D&D allows for the 20 - 100 HP range, I think, but what they stand for, no one knows!
Believe it or not, I actually think that is a great idea, and in Nexia, the amount of HP a character has isn't very important. In fact, it is randomly generated at creation (20 + [4d6 - lowest die]). There will be a way to gain more points through training, but I expect many characters to not pay much attention to that.
However, it will diverge from the D&D system in how the abstraction lends itself to other mechanics in the game.
For instance, the keyword allows for damage types. This makes sense at first, since if I am hitting something with a flaming sword, it should do fire damage. They have rules for resistance and vulnerability, which play in strongly for certain class builds.
This issue with that is the rules get relatively complicated when multiple keywords are applied to an attack. How does someone with fire vulnerability and necrotic resistance deal with a Fire and Necrotic attack?
There is a way to resolve it, but why? We don't even know what hit points are in D&D. How much is represented by the mental capacity of the character, as opposed to the physical ability? And what does magic protection do? Some of these are answered by the defense that is attacked, which is a neat aspect of D&D 4e; however, we don't have a clear idea of what HP represent.
In Nexia, mechanics are balanced between abstraction and simplicity. HP is just a convenient way to track damage. It isn't the focus of the game, except "on the grid", and then it is really an attendance bar: depending on the enemy, you have these many points before you are out of the engagement.
We've moved the vulnerability/resistance mechanic to the roll itself (expressed as Fire Resist 2, that means you subtract two dice from the roll against you, either not having them rolled, or the highest and lowest, in group attacks). In case there are ever keywords that contradict each other, you do the operation before the roll (Necrotic/Fire attack against Fire Resist 2 and Necrotic Vuln. 3 becomes Vuln. 1).
I doubt there will be many cases where that comes up (well, never, actually, we aren't using a Necrotic keyword), but I wanted to share the design policy in abstracting rules. Our goal is to create a game that focuses on team play and fast resolution. To that end, I look at each aspect and see how we can make it more fun, at least not make the nonsense a game-stopper.