It is true in a Heller triangulated category (or $\infty$-triangulated category - strictly speaking one only needs a 3-triangulation) 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 is part of the axiomatics).

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 Theorem 2 in Maltsiniotis' preprint).

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

It is true in a Heller triangulated category aka $\infty$-triangulated category (although strictly speaking one only needs a 3-triangulation for octahedra) 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 is part of the axiomatics).

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 Theorem 2 in Maltsiniotis' preprint).

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

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

It is true in a Heller triangulated category (or $\infty$-triangulated category - strictly speaking one only needs a 3-triangulation) 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 is part of the axiomatics).

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 Theorem 2 in Maltsiniotis' preprint).

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