Death Save Recuperation

When playing D&D, we often run into the situation where a character goes down in combat, gets healed back up, immediately begins fighting, goes down again, and the cycle repeats. While this fits the rules, it negatively influences the fiction of the game. We don't expect someone to be stabbed in the chest and nearly die only to stand back up and begin fighting with no change, it breaks our immersion in the game world. In order to fix this we don't actually have to do that much, the result not affecting the standard gameplay almost at all, but influencing a players mentality greatly. 

Death Save Recuperation

When you suffer a death saving throw failure, that failure lasts until it is removed with a medicine check, spell usage, or other type of healing and a day of rest. If you go unconscious while you already have a death saving throw failure, that failure carries over to your current death saving throws.

Example:

My character Aragorn suffered 2 death saving throw failures after he fell off a cliff, but has miraculously survived. During his journey to rejoin the party he stops at a village tavern for a day to recuperate from his wounds, and the village doctor comes and performs some healing, removing one of his death saving throw failures. The next day he rejoins the party, and during a battle against a horde of orcs he gets knocked down again, and since he had one death saving throw failure remaining from when he fell off the cliff, he now automatically has one death saving throw failure. Luckily he gets healed back up before he has to roll another death saving throw, but his past experiences are clearly wearing on him and he needs to be careful.