What if HP is the total of all the damage you have taken which acts as the DC you need to beat when you take damage with a D20 roll.  You add a modifier based on the classes hit die (see below).  The idea is that casters (Wizards and Sorcerers) can't add there entire proficiency so there still squishy.   If you fail three times your down and you start makin death saves.  Each time you fail, take a long rest, or on a natural 20 your HP resets to 0.  Healing (hit die or otherwise) now subtracts from your HP so you have a easier time making the roll.  Temporary hit points act as a buffer, so you can take that much damage without increasing your HP, you still need to roll though.  Not sure if you can have negative HP.. maybe with a magic item or something.

NOTE: I wouldn't roll damage if I did this but would instead use average damage so two people aren't rolling, only one.  Definitely works better on low levels than high.  Still thinking on how to fix that.  I'm kinda looking at the ancient black dragon breath for comparison.  I might even have to change damage to have a max, so instead of it's breath weapon doing 67 it does 16.  Assuming your proficinecy is +6 and you have at least a +1 in constitiution your would need to roll higher than a 9 to not take a failure on a hit which seems fair.  So enemies would only do damage less than 16 damage ever.  I don't like having to change every monsters damage.  Again still working on it.

Now, why would I do this.  It does make HP more than just a number that can hit 0.  Now HP is a mechanic that can now be played around with by feats, magic items, home brew classes.  One advantage and disadvantage (at the same time) is that now there's no set amount of damage a player can take before going down.   A fighter can go down in three hits if they roll three one's.  On one hand, that sucks, on another it's a hell of a story when your big fighter got knocked down by three tiny goblins that wouldn't have hurt that much using normal HP.  Alternately a wizard can just keep on getting natural 20's and take way more damage then they were ever supposed to be able to take.  It also keeps enemies dangerous, even if they normally wouldn't be a challenge you can always roll shitty and go down.  It kind of reminds of dark souls games.  Even level 1 enemies can kill a high level character if they hit you with a frensy attack, this is a similar deal.  The enemies might only do a d4 damage but your normally 40 HP fighter could still go down to three fails. 

Not to be mean, but I have a lot of questions/problems with this from both a mechanics and game experience standpoint

1) I really feel like you are making what should be the easiest thing to keep track of in game (when there are a lot of more complicated things to track already) a lot harder to track.  

2) assuming this gets applied to monsters as well, you   Are completely taking the players accomplishment away from combat.  If everything comes down to saves,  then your character is not rolling to hit, ever, which for me takes a lot of fun out of combat as the player never gets the satisfaction of a high roll, or a crit.  It also lowers the feeling of accomplishment if a level one party could kill a CR 20 dragon that crit failed a saving throw, and then at level 15 be killed by a group of goblins when the party fails the saves

3) the concept of a high level fighter going down in three hits from a goblin might sound like an interesting story, but I’m sure the player running that fighter won’t see it that way

4) how does this scale?  In the current game the HP difference from level 1 to 20 is huge, representing a significant gain in power.  Your mechanic seems very limited in its ability to scale up.

Not applied to monsters.  Never.  Working on the scaling.  Just an idea.  PS Mutants and Masterminds already basically work like this.  They call it a toughness check.