About the cone being unique up to non-unique isomorphism 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. 
 A: Another theory (equivalent, I believe, to derivators) which has the aim of compensating for this shortcoming of triangulated categories is the theory of ∞-categories.
These are, essentially, categories where you have a well-behaved notion of (higher) homotopies between morphisms. As such it is unsurprising that they have found fruitful applications in homotopy theory, but they are also useful in more classical algebraic geometry. For example they are a useful framework in which to study derived algebraic geometry, with applications to the Langlands program.
Pretty much the idea is to get back the uniqueness of the mapping cone by remembering the nullhomotopy of the composition $X\xrightarrow{f} Y\to Cf$.
A: As a first idea, not only the cone but the so-called higher homotopy constructions, like the cylinder, homotopy (co)limits, etc. are not well suited to the derived category as a bare triangulated category. A way of dealing with all these issues and at the same time re-inventing a part of homotopy theory of derivators. This was developed by Grothendieck during the years 1990-91 but never published it. It was unearthed by Maltsiniotis and Cisinski around the beginning of the millennium. There is a page with Grothendieck's original writings and further texts developing this circle of ideas. It is
https://webusers.imj-prg.fr/~georges.maltsiniotis/groth/Derivateurs.html
A nice and direct introduction is Groth's 
"Derivators, pointed derivators, and stable derivators", 
Algebr. Geom. Topol. 13, no. 1, pp. 313–374 (2013).
Also mentioned in this page. You can find it on arXiv:
https://arxiv.org/abs/1112.3840
