Star Trek went into this a bit. There was an episode in which the technology behind the android Data's artificial brain had been turned into a medical treatment used to prolong the life of a recurring character who got injured, and the ethical question of the subplot was roughly "how much of their brain can you subsitute with functional but essentially generic hardware before they cease to be themselves, making the rescue attempt fruitless".

Most people will probably not remember this - it was the death of Kira's lover Bareil on DS9 - because it was almost throwaway, but it was one of those sparkling little ideas and questions Star Trek used to be filled with that has stayed with me life-long.