A word to refer to the sound that an elevator makes when its cables are having problems?

I suggest screeching:

: a high shrill piercing cry usually expressing pain or terror

I'd use graunch, which is rather onomatopoeic. And would you believe it, it's in the dictionary, even though my spell-checker doesn't like it.

graunch verb
[no object]
make a crunching or grinding noise:
the wheels graunched against a stone wall

Onomatopoeias are very subjective, and depend entirely on the sound, to say nothing of the different interpretations in different languages for the same sound. So bear in mind that any answer given is going to be subjective to one's own experience.

You might get a "creaking" noise, if the elevator is sticking and the metal parts are trying, but failing, to move against one another. Or if the metal elevator is straining against the cables, or if the wire is being stretched.

"Creaking" works especially well, since it includes both the squeaking sound one might hear from old, rusty, unoiled cables, as well as the grating noise one would hear from the metal parts pushing and grinding against one another. Even moreso, since it also describes an object that is moving whilst making this sound (A creaking elevator could mean that it is making this sound while moving, both implying the sound and the difficulty with which it moves).

Screech is the word you seem to be referring to when a car stops abruptly ( screech to a halt) which could also apply to the elevator cables under some circumstances I suppose.