The Beauty of Hit Points
Hit points in D&D have a long and storied history for a reason: they are a superb way to get players feeling the tension of the game. Human beings are loss-averse, so the whittling away of hit points to 0 makes them feel like they are loosing something special. The fact that 0 hp creates unconsciousness and begins the chance for death is also key: there is a penalty for loosing all of this resource.
So HP, as a game mechanic, work really well. It hits the players' fear, helping to represent the character's fears. They're a genius little design trick.
The Ugly of Hit Points
Just because HP make good game mechanics doesn't mean they're problem-free. People have problems with the "realism" of HP, specifically their binary nature (how, at 1 hp, you're fine, and at 0 hp, you're dying).
Another big complication comes in that HP escalate with level, and what would kill a 1st-level character might not kill a 10th-level character. Sure, characters are "tougher" as they gain levels, but a PC that can survive a terminal velocity fall into a pool of magma (example from my own campaigns) isn't the kind of fantasy RPG that everyone likes. It's clearly superhuman. If you like your high-level characters to be like Achilles or Gilgamesh or Hercules, this isn't a problem. If you like your high-level characters to be like Aragorn, it is.
There's also problems related to how HP is recovered without magic: rest healing your wounds makes sense, but how long does it take to get back up?
And another problem, with the kinds of things that deal damage. If a housecat's claws deal damage, then farmers (or low-level wizards) can be killed by them. That appears on the other side of the coin from "surviving falling damage:" something fairly weak and inoffensive might end up being deadly.
A Brief View of Fixes
Most of the time, HP have been fixed by creating a separate system for "real" damage. Whether it is a "Death Spiral" mechanic (perhaps similar to a 4e disease track) or a "Wound Point" mechanic (featured in 3e's Unearthed Arcana, and in the Star Wars d20 games) or a "Healing Surge" mechanic (a la 4e). These fixes work OK, though they add some complexity. What their real problem is is that they lack the psychology of loss-aversion present in a pure HP system. With wound points and death spirals, your fate feels less in your hands, more removed from the obstacles you face on a regular basis. It becomes a more rarely-used mechanic, something used only for the special circumstance of ACTUAL damage. This keeps HP as a pure plot-armor device, and relegates "actual danger" to some secondary subsystem. The threat is no longer front-and-center: a sword attack probably won't hurt you too much. It deals HP damage, which is at least one degree removed from the kind of damage that kills you.
It also tends to be a problem that the complexity introduces unnecessary realism. In D&D, a game of heroic fantasy, I don't want a sprained ankle or a broken leg to handicap my fighting ability. It's realistic, but it's not fun action fantasy, it's another fiddly bit to track. A wound system or a death spiral mechanic introduces that complexity and that realism where I don't necessarily want it. I might be fine with HP representing wounds.
From The Ground Up
In looking at how to parse this problem, I wanted to look at what you would HP need to do, in the absence of any other system. That is: what is the basic function of HP in play, and what follows from that basic function. What is the minimum possible thing that HP can do in the game.
It tuns out that this is Principle One:
Principle #1: Hit Points Are For Not Dying
So, to keep the D&D-ish-ness of HP, this principle is inviolate: Your character has some quantity of HP, and, when it runs out, they are at risk of dying. This, I think, is the basic idea of of hit points: they are there to prevent you from dying in one hit.
Principle #2: Anything That Deals HP Damage Can Kill You
Because hit points are for not dying, anything that deals HP damage is something that is potentially lethal in some way. Sword hits, falls, badger teeth…but probably not pixie swords, individual horsefly bites, and cat-scratches. Poison that is deadly yes, poison that causes some mild indigestion, no. A physical blast of magic, yes, a demoralizing mental attack, no. This means that HP represents your skill at not dying from something that would be lethal to a less skilled individual: you're tougher, more resilient, and more rugged. The things that do not do HP damage may still affect you in some way, but they don't need to reduce your HP total.
Principle #3: Anything That Heals HP Damage Can Undo Death
Because hit points are not for dying, and because anything that deals HP damage is potentially fatal, when you heal HP, you need to be able to prevent death. Thus, divine magic, yes. Resting…well, for heroes, yes, but over long periods of time. Herbs and poultices, bandages and medicines, of course (this would be the "Martial Healing"). Yelling at someone? Restoring their morale? Giving them a pep talk? Not so much. The things that do not heal HP damage may still affect you in some way, but they don't need to increase your HP total.
Principle #4: We Still Want Fate
HP in the earliest models were a mix of actual wounds and more esoteric luck, fate, grit, and divine favor. This is a key point of the abstraction, and helps DMs maintain their own degree of believability in the world. While HP must represent some physical wounds, a character who takes ten sword hits and remains standing can be too "wahoo" for some styles of the game. A character who takes ten "near misses" from swords, but who hasn't taken much in the way of a phyiscal injury, can help maintain that believability. We want to keep that element.
Principle #5: The Cleric Must Be Optional
As much as HP must be healed with things that can heal actual wounds, we also don't want to require the party to use divine magic to be operational. We want a party of all thieves, or a party with an extra wizard or a party in a world without magic to all work just fine.
Where We Are Now
The 5e playtest currently assumes that your HP is 99% luck, skill, and divine favor. Only your very last hit point — the one that brings you from 1 hp to 0 hp — is an actual injury. The documents describe any hits when you're below half to be cuts and bruises and scrapes, and any hits when you're above that to be near-misses, unbalancing impacts, and ankle twists, but none of those are significant injuries. So a hypothetical 5e inspirational healer like the 4e Warlord would work fine, as long as the person wasn't unconscious. This is why you gain all your HP back with a night's rest, and why you can gain some HP back with a five-minute break. It's also why it takes longer to recover if you're at 0 hp: 2d6 hours before you can take an extended rest might mean it takes an extended rest to get you to 1 hp, and then you need to take ANOTHER one to recover your resources.
This hits all the principles well enough. By the terms laid out here, every character has 1 hp, and then the rest of their "hit points" are actually "fate." This makes some problems for those that want HP to be more physical. It also creates some rules-weirdness. HP in this model shouldn't get a bonus from your CON score, it should get a bonus from your DEX score, or your CHA score! Maybe you add CHA to your attack damage instead of STR. AC doesn't represent your ability to avoid attacks, it represents the attacker's ability to get a "close call."
These issues can be abrogated by altering the percentage of your HP total that is "actual" HP. We don't want it all to be injuries, in the end: that makes it difficult for healing other than divine magic to work.
Easy Fix #1: Your CON score is your "actual hit points."
You can keep everything as it is. Just have the players note when their HP drops below their CON score, and at that point, narrate attacks as actually injuring the character. When you take a short rest, you can't use your HD roll to heal this damage. When you take an extended rest, you don't automatically heal these points (you heal up to your max HP - your CON score): you need to take a Long Rest (one week) in order to get these back. If you want to make this distinction clearer, you can label your CON score your actual Hit Points, and everything above that Fate.
Easy Fix #2: Half Your Hit Points are "Actual Hit Points."
You can keep everything just as it is, but, as above, when your players drop below half their HP, begin to narrate attacks as actual injury (even if it's not a bad injury). When you take a short rest, you can't use your HD roll to heal this damage. When you take an extended rest, you don't automatically heal these points (you heal up to half your max HP): you need to take a Long Rest (one week) in order to get these back. If you want to make this distinction clearer, you can label half your hit points as your actual Hit Points, and everything above that Fate.
Easy Fix #3: 0 HP Is Harder To Recover From
The "2d6 hours" rule is a little random and weird, but it's nice to have 0 hp being a "real" injury. Instead, 0 hp might take an extended rest to heal (so that you're at 1 hp after one extended rest).