Maybe this is not what you would like but it is true in a Heller triangulated category (or $\infty$-triangulated category) that any morphism between the bases of octahedra (by which I mean the 3 objects and two composable morphisms from which the octahedron is built) extends to a morphism between the octahedra.

It turns out that the triangulated categories which turn up in "nature" all satisfy these stronger axioms. For instance the homotopy category of a stable model category is $\infty$-triangulated (a more general statement holds which is proved in Maltsiniotis' preprint).

References are M. Künzer. Heller triangulated categories and G. Maltsiniotis. Catégories Triangulées Supérieures