Here's what I'm putting in my houserules document.
Each time you fail a death saving throw (which occurs when dying), you must roll on the following table (derived from the lingering wounds system on page 272 of the Dungeon Master’s Guide).
d6 d20 Effect
1-3 - No Effect.
4-5 Minor Wound.
1-2 Minor Scar. The scar doesn't have any adverse effect. Magical healing of 6th level or higher, such as heal and regenerate, removes the scar.
3-5 Limp. Your speed on foot is reduced by 5 feet. You must make a DC 10 Dexterity saving throw after using the Dash action. If you fail the save, you fall prone. Magical healing removes the limp.
6-8 Strained Arm or Hand. You can’t use either your primary or secondary hand (determine randomly); you can no longer hold anything with two hands. If you are forced to make attacks or ability checks using your secondary hand, you have disadvantage. Magical healing removes the strain.
9-11 Broken Ribs. Whenever you attempt an action in combat, you must make a DC 10 Constitution saving throw. On a failed save, you lose your action and can't use reactions until the start of your next turn. The injury heals if you receive magical healing or if you spend ten days doing nothing but resting.
12-13 Internal Injury. This has the same effect as Internal Injury above, except that the save DC is 15.
14-15 Festering Wound. Your hit point maximum is reduced by 1 every 24 hours the wound persists. If your hit point maximum drops to 0, you die. The wound heals if you receive magical healing. Alternatively, someone can tend to the wound and make a DC 15 Wisdom (Medicine) check once every 24 hours. After ten successes, the wound heals.
16-17 Staggered. For one minute, you have disadvantage on attack rolls, ability checks, and saving throws, and cannot maintain concentration on any spells. The condition is removed if you receive magical healing.
18-19 Weapon or Tool Damaged. Whatever you were holding in your hands is unusable (determine randomly if you are holding more than one item). You may be able to have the item repaired by a competent craftsman.
20 Armor Damaged. Your armor is unwearable. Until you can remove your armor (see page 146 of Player’s Handbook), you gain the restrained condition, except that your speed is halved not 0. You may be able to have your armor repaired by a competent armorer.
6 Major Wound.
1-3 Lose an Arm or a Hand. You can’t use either your primary or secondary hand (determine randomly); you can no longer hold anything with two hands. If you are forced to make attacks or ability checks using your secondary hand, you have disadvantage. Magic such as the regenerate spell can restore the lost appendage.
4-6 Lose a Foot or Leg. Your speed on foot is halved, and you must use a cane or crutch to move unless you have a peg leg or other prosthesis. You fall prone after using the Dash action. You have disadvantage on Dexterity checks made to balance. Magic such as the regenerate spell can restore the lost appendage.
7-8 Lose an Eye. You have disadvantage on Wisdom (Perception) checks that rely on sight and on ranged attack rolls. Magic such as the regenerate spell can restore the lost eye. If you have no eyes left after sustaining this injury, you're blinded.
9-11 Horrible Scar. You are disfigured to the extent that the wound can't be easily concealed. You have disadvantage on Charisma (Persuasion) checks and advantage on Charisma (Intimidation) checks. Magical healing of 6th level or higher, such as heal and regenerate, removes the scar.
12-14 Sapped Vitality. You always have at least 1 level of exhaustion that cannot be removed until you spend ten days doing nothing but resting or you receive magical healing of 6th level or higher, such as heal and regenerate.
15-16 Concussion. You are unconscious for 1d4 hours (even after receiving healing that would normally wake you). For 24 hours, you have disadvantage on attack rolls, ability checks, and saving throws, and cannot maintain concentration on any spells. Magical healing of 6th level or higher, such as heal and regenerate, removes this effect.
17-18 Broken Jaw. Your jaw is broken so that it interferes with your speech. You have disadvantage on Charisma (Persuasion) checks. It takes you ten days to learn to cope with the injury; during this time, any spells with a verbal component are 50% likely to be miscast (you spend the slot but there is no effect), and your speech is slow and laborious. Magical healing of 6th level or higher, such as heal and regenerate, heals the injury completely.
19-20 Psychological Trauma. You gain the flaw "I am afraid of [the situation or creature that almost killed me]." In similar situations, the DM may rule that you have the frightened condition. With the DM's permission, you can have an alternate flaw that represents your trauma (see Madness in the Dungeon Master's Guide). Calm emotions can suppress this trauma, while greater restoration or similar powerful magic can heal the trauma completely.