A realistic health system in the sense that if someone shoots or slashes into both your legs, you fall. And probably fall unconscious, from the pain. There's a chance you survive though. So a difference between down and dead, basically, but a realistic one. And if you get shot/stabbed in both your hands, you most likely drop your gun, right? Some wounds recover, some cost you your legs... Some need to be dealt with immediately or the impact remains...
Those wounds that remain should remain throughout the game, even when you finished the mission.

It would most ideally be some squad game so unconscious isn't game over since you've got other members to save you and such.

I don't care much about the timing. Perhaps having it involve money somehow (hiring a crew, buying guns...) would be nice.

Just name everything that comes to mind I suppose.