5e Death Save Alternatives: Two Different Ones

Type I: Brutal Wounds 

When you get knocked down to 0 hp (assuming lethal intent), you are wounded, and have a chance of dying. Roll 1d20 to see how bad it is:

20: But a scratch. Oh, I guess you weren't hurt after all. You're at 1 hp.

15-19: Flesh wound. Disadvantage on ability checks until short rest.

10-14: Serious wound. You fall prone until someone helps you up. Disadvantage on attacks, checks, and saves, Speed is reduced by half. If you have access to spells such as cure wounds, the wound can be healed over a long rest. If you don't, it could take months.

5-9: Death's door. Unconscious. Make another save on your next turn:

20: you wake up with 0 hp and a serious wound

15-19: stable; wake up in 1d4 hours with 0 hp and a serious wound

5-14: still at death's door; try again next round

1-4: death

Any further attacks while at death's door are lethal.

2-4: Death (heroic/tragic/dramatic).

1: Death (comedic/embarrassing).

If you're lucky enough to survive at 0 hp, any further damage you take will call for more death saves. Roll with disadvantage, and ignore any result that's not worse than your current situation.

All glory to The One Death and Dismemberment Table to Rule Them All, but it was a bit too complicated for my tastes (and potentially too graphic for general audiences). My version is simpler and more handwavy, though it shares some of the inherent problems (like interactions with healing mechanics, which will have an impact on the structure of the campaign).

Type II: Creeping Death

When you are reduced to 0 hit points, you fall unconscious and are dying. Roll a death saving throw immediately, then at the start of each of your turns:

20: Two successes
10-19: Success
2-9: Failure
1: Two failures

Keep track of your successes and failures. When you get three failures, you die. When you get three successes, you stabilize. When you are restored to positive hit points, you stabilize.

When you stabilize, erase your successes and stop making death saves. You wake up after about an hour. Do not erase your failures until after you finish a long rest.

The goal with this system was to create a lingering consequence for falling to 0 hit points, but without crippling the character. In this system, your death save failures pile up over the course of the adventuring day. So, each time you fall to 0, you have a chance of sliding closer to death. 

In vanilla 4e and 5e, the dominant strategy for healers was to wait until someone got to 0, then heal them. I've heard it called "jack-in-the-box combat." He's down! He's up again! He's down again! He's up again! Not so in this system.