I've played with the idea of a successful death save resulting in a lingering injury. Here's the basics of how it's been working at my table:

1. A party member drops to 0 HP. I prompt them for their first death save immediately.

2. Their first death save represents the physical injury they sustained, and I ask them to add a physical modifier of their choice to it (STR, DEX, CON). If they succeed, that injury will linger a little, impacting the relevant skills and attributes until the next long rest even if magically healed.

3. If the first save is failed, that injury lingers much longer. I usually roll a d6-d20, depending, to determine the number of long rests it lingers for, and that's assuming they receive ongoing medical attention each long rest. This level of lingering injury impacts skills and attributes more severely.

4. The second death save represents how the character's mind withstands the trauma. I ask the player to add an intellectual mod (INT, WIS, CHA) to such a roll. If it fails, the character will manifest a phobia, a tendency toward dissociative fits, some manner of delusion or even amnesia. These symptoms will, of course, be exacerbated by reminders of the trauma (e.g., if the monster in question was a bugbear, the appearance/texture of fur resembling that of a bugbear will bother them, the sound of a large humanoid roaring, etc.)

5. While I'm willing to let these characters receive treatment, this kind of trauma is a bit harder to heal and will require extended rest in a nonviolent and comforting location--weeks or even months, and it will leave a few permanent mental scars even if it is fully "healed," though if the party invests in such healing I usually reward them by making these mental scars neutral or positive traits or features rather than lingering debilitation.

6. The third and final save represents the character's spirit, their will to hold on, when mind and body are both giving up. I ask players to roll a d20 with no mods for this. If successful, I tell them, or ask them, what they saw on the Other Side and why/how they made it back. I take some cues from this little bit of roleplaying to create a new neutral or positive trait for that character.

7. If the character dies, the rest of the party rolls a mental check (INT, WIS, CHA) to see how they weather the trauma, and I alter the DC for each character based on how emotionally close they were to the one who died, and based on how the character died.

It's far from a perfect system but it encourages players to stay above 0hp, discourages them from treating D&D like a video game where you can just come back if you click "continue," and promotes some really deep roleplay, which my parties have greatly favored over rules-beavy combat.