In an answer to this MO question [link] Fernando Muro sais:

the mapping cone of a morphism in a triangulated category is unique up to non-unique isomorphism. This fact has originated a lot of research in this topic, and it still does.

I would be curious to know (not necessarily from the author of that answer himself!) about the research, past and present, for which it could be said that it has more or less directly originated from the above observation.