I do like the rule someone mentioned where failed death saves carry over and only clear on a long rest, it's one I've considered implementing but ultimately decided not to, at least for the current game. If I want to run a game where death is a greater threat I'd use this rule and, to make it even scarier, each time you hit 0 you automatically gain a failed death save.
I think currently we're using a scar system which has 0 effect on gameplay and is really only up to the player to keep track of for fun. Each time you hit 0, you gain a scar and make a note about how you almost died. If we were going to go all in on this system, then I might make a system where scars grant bonuses and penalties to certain checks.