The 5e Dungeon Master's Guide contains an optional rule on p.272:
A creature might sustain a lingering injury under the following circumstances:
- When it takes a critical hit
- When it drops to 0 hit points but isn't killed outright
- When it fails a death saving throw by 5 or more.
When an injury happens, the player has to rule a d20, and there is a table. Some examples are
- lose an eye
- lose an arm or hand
- lose a foot or leg
- broken ribs
- minor scar
The generally have some suitable negative mechanical effects attached (no two-handed weapons for a single-handed warrior). Most of the injuries can be healed using magic, but some of the worse ones require a high level spells such as L7 regenerate. This means that removing the injury can be integrated into the story, especially for lower level parties, as they would need some higher level NPC to heal them.
We used the lingering injuries rule on crits and death saves in our last session, and they were great. They make crits memorable, and create fear and despair in players similar to the older editions death and dismemberment table. I can only recommend you to use these rules.
If your DM only wants to use a less severe table, he is of course free to create one himself. Googling for 'death and dismemberment' should find plenty of options in various forms of severity. A table I used for an earlier campaign can be found here.