In 5th Edition hitpoints are defined as combat stamina and luck. A character doesn't get healthier by gaining levels, he just gets more accostumed to fighting and pain, and more experienced at avoiding lethal blows. This is why a character can "heal" overnight from 1 hitpoint, and this is why short rests make sense. Hitpoints do not represent actual physical health and well being, and for this reason hitpoints only make sense in combat situations. With these variant rules, you can implement a more realistic (and grittier) way of treating non-combat damage.
1. Non-combat damage:
Adventurers do all kinds of dangerous things apart from fighting and they take damage all the time from traps, falling down from somewhere, poison and other nasty things. With this rule you can make these sources of injuries WAY more dangerous and lethal. Make any non combat damage source deal Constitution damage instead of hitpoint damage. Not the same amount of course, as a general rule I use, every dice of damage deals 1 point of CON damage instead. Therefore 5d6 falling damage would turn into 5 CON damage. Since the damage dice are different for many sources of damage you can convert everything into d6-s easily, or take the average damage and divide it by 4, rounding up or down as you feel like.
Now this will make your traps and other non-combat encounters very dangerous and unforgiving. No longer will a trap only take some HP that the party can regain with a short rest, but stepping into a trap could be fatal, or very close to it, and can easily end a character, or a whole party (as they were designed) so use them sparingly. Or don't. You are the GM.
You can apply this kind of damage to falling damage, fire damage from a non-spell fire, falling into acid or really anything. You may or may not want to use this for your traps as well. I personally barely use traps, but when I do, I make them count. If you like to use traps a lot, don't apply this rule to traps.
2. Coup de grâce and helpless opponents:
Normally, when someone is helpless, you can just kill them outright, and not "attack them with an advantage", or have an automatic critical strike with sneak attack etc. Cutting a someone's throat while they are sleeping isn't rocket science, and you don't need to take 10 levels of rogue before it becoming effective.
I usually use one of two different ways of handling attacking helpless opponents.
A, When the party is in combat or under stress, but they managed to make one of their opponents helpless, I use this: The attacker rolls the attack roll with an advantage and deals damage normally, but instead of hitpoints, it deals CON damage. The full damage. This is likely to kill the enemy outright, or cripple them for life, but the attacker can also miss by 1 or 2 points of Con, leaving the NPC alive (sort of). But they do not know that, and the enemy can come back later. MWUHAHAHA. (Warning: This makes Hold person/Monster REALLY strong!)
B, When the enemy is completely helpless, and the attacker has the chance to prepare for the strike without disturbance, I just roll a d20 to see how lucky is the victim. On anything else but a 20 the player manages to kill the opponent, with a 20, they can survive somehow (the player is not aware).
I know that balance wise none of these rules make sense, but common sense trumps game balance. Of course, I do not allow any of these rules on creatures that cannot be slain like this. A dragon could be hit with a dagger while he is asleep for example, but it wont damage him at all.
3. Instant death:
In some cases instant death is inevitable. Falling several hundred feet, drinking lava, playing hide-n-seek in a dwarvish smelter usually causes things to die. In some cases there is just no room for rolling at all. Getting chomped by a dragon is deadly, no matter what armor you are wearing. Whenever I can, I try to avoid PC deaths like this, but as I mentioned, common sense trumps rules.
If you want to use these rules, but you think that healing constitution damage is too hard to acquire, you can use one of the following rules: (the CON healing only applies to damage taken from the above sources)
You can heal 1 point of CON damage per healing magic used on you.
You can heal 1 point of CON damage per long rest.
You can heal 1 point of CON damage per week.
//Some of these houserules were inspired by The Legendary Gothnog, and his games I played in//