Traditional Abstract HPs
Hit points are a mix of physical toughness, health, will, luck, divine favor, magical protection and so forth. Hp damage can be just about anything, from a bleeding wound to exhaustion to, to psychic assault - any attack or environmental factor that can bring you closer to death or unconsciousness can be hp damage. By the same token, anything that can help with any of those factors that make up hit points can heal.
Don't worry about it, just track hp loss and restoration.
Possible 5e Mechanics: The current 5e system works fine for this, and any attack or healing could be added with no issue to speak of. Just don't bother differentiating types of hps damage vs ways in which it's healed.
Derived from E.Garry Gygax's monographs on hit points in the DMG 1, but taking them a bit farther, this option assumes that characters start out with 'meat' hit points (physical structure's resistance to being destroyed), and gain health from CON, and other factors (as in the traditional system) from leveling. Thus, a character of any apreciable level has a ratio between his 'real' hit points and his 'non-physical' hit points. That ratio is respected in both damage-dealing and healing. So, if a 100 hp character takes 10 hps of damage, he's wounded to exactly the same physical degree as if a 10 hp character took 1 hp of damage. Thus, the ability to heal a 1hp wound for the 10hp character should be able to heal the same (10 hp) wound on a 100 hp character.
There are two possible ways to handle this: 1) allow only forms of healing that actual make wounds go away, and have them restore non-physical hps in proportion to the wounds they remove. 2)Track real and non-physical damage separately and have different types of healing restore one or the other.
Possible 5e Mechanics: 1) Powers that actually heal (restore physical damage) heal amounts in proportion to the healed characters total hps. Very minor healing heals 1 hp, regardless, that's always the minimum. Light wounds, up to 1/4 hps can be healed by apropriate magic, more powerful magic heals more. Rest and time heals at a rate of 1/50th hp total/day, minimum 1 (ie, it takes 7 weeks to heal completely from near-death). That rate could be varied to suit the 'tone' of the game with just a little math. 2) Characters have physical hps equal to their CON + level, even if this is more than their total hps (this just means that a low-hp PC can be dropped or killed more easily than dismembered), the balance is non-physical hps that allow him to avoid physical damage. All damage, physical or non-physical in nature though it may be, comes off non-physical hps (if any) first, and is able to inflict physical hp damage thereafter. Healing must also be either physical or non-physical and only restores the corresponding type of hps. A character brought back from 0 by physical healing is still realing/demoralized/out-of-luck/whatever, and all damage he takes will be physical. A character brought back from 0 by non-physical healing is still wounded, he's just ignoring the wound (and able to avoid others) through sheer determination/luck/etc. A character healed beyond his non-physical hp maximum by non-physical healing gains temporary hps equal to the difference. These extra hps are taken first and, if any are left, disapear after the end of an encounter. Excess healing of physical hps has no effect. A character who rests 8 hours regains all his non-physical hps. Healing physical damage through rest & time requires convalescence lasting 7 weeks less one day per level or point of CON modifier.
Life Energy Economy
Derived from the actual mechanics of classic D&D and a bit of reading between the lines, this mystical aproach to hit points assigns all creatures hit points that represent their capacity to hold life energy. Life energy is that mystical power that separates the living from the inanimate. Just as the elements of which the physcial world is composed come from the elemental planes, life energy comes from the positive material plane, and the antithetical energy of undeath from the negative material plane. All living creatures have life energy in proportion to their pysical size and health, but many have the capacity for additional life energy as a result of magic, divine favor, determination and experience. When a living thing is hurt, the damage to its physical form expels life energy, and it is this, not the mere physical damage, that actually kills. If it has a surfeit of life energy, a creature can sustain multiple "lethal" wounds without even slowing down. All living things receive a slow trickle of life energy from the positive material plane in proportion to their size and vigour which they use to live & be active, when resting, this life energy accumulates, allowing them to heal injuries. In addition, magic can evoke energy from the prositive material to restore life energy. In both cases, when life energy is restored, any physical damage that caused it to be lost is, likewise, repaired.
Possible 5e Mechanics: Characters gain hit points from class/level and CON (as currently). Smaller characters still have the same life energy, for the sake of fairness to gnomes and halflings (optionally, they can have a small hp penalty). Only magic can heal quickly by instantly infusing the subject with life energy, magical healing thus simply restores hps and closes any associated wounds instantly. Excess healing has no effect, at most it can 'top off' the character's maximum store of life energy, but other kinds of magic an infuse a being with extra life energy for a time. Healing via rest and time is a slow process. Living beings of PC size (from the tiniest gnome to the burliest goliath) heal 1 hp per day of complete rest, plus their CON bonus per week whether they rest or not (optionally, small PC heal 1 less hp per week and large ones 1 more).
Cinematic "Plot Armor"
Hit points represent the ability to avoid and mitigate damage by any means, from sheer toughness ("soaking up" a blow without really being hurt), to speed and skill, to improbable luck, divine grace, or whatever. Until a character is out of hps, none of the 'hits' he take inflict meaningful physical (or any other sort of) damage. He might get scratched or 'shot in the shoulder' or something. Once a character is out of hit points he /may/ take a genuinely dangerous wound, but the "audience" doesn't know until its resolved. A PC impaled by a spear may not have suffered damage to any vital organs, or he might have been pierced through the heart - he might be up and moving about in moments or dead before he hits the ground, you don't /know/ until the system for dying has been resolved. As long as a character has hps, he doesn't suffer any lethal fate, so, for instance, taking poison damage means he's avoided being poisoned (or at least avoided a fatal dose).
Possible 5e Mechanics: Characters have hps based on class/level & CON. Characters also have a 'recovery' stat, a number of HD plus a bonus, that yields a maximum equal to half thier hps. In combat, the dice are rolled when the character recovers. Out of combat, he can take the maximum for each recovery. A character has a limitted number of recoveries per day, based on class, CON, and the tone and/or pace of the campaign. Recoveries can be taken once in an encounter as an action, or more if he has a special ability to do so, or another character has such an ability to grant others recoveries. Between combats, any number of remaining recoveries can be used to restore hps.
When a character is reduced to less than half his hit points, he takes a visible (but not debilitating) wound represented by the "Bloodied" condition. A character reduced to 0 hps is "dropped" and unable to act, but his condition apart from that is unknown. Once dropped, a character must make Death Saves to resolve the extent of his injuries. A failed death save indicates at least a serious wound. A second death save indicates at least a critical wound. A third failed death save and the character is dead.
Healing removes the Bloodied condition and failed death saves (which otherwise persist even if a dropped character is brought back to positive hps). Cure Light Wounds removes the bloodied condition, Cure Serious removes a failed death save (if only one has failed), and Cure Critical removes two from a living subject. The Heal skill can suspend the bloodied condition or one or two failed death saves, with a good enough check result (progressively harder the more times it's been done), but does not cause the wounds to disapear, merely 'stabilizes' them so they are not a problem. Rest & Time removes the bloodied condition in an extended rest, or a short rest with a successful endurance check. Failed death saves are removed at one per full day (or week or month depending on campaign tone) of complete rest, or per extended rest with a successful hard endurance check.