For a given orientable $n$-manifold, is there a canonical choice of $n+1$-manifold that has this manifold as its boundary?
For $n< 3$, every orientable manifold can be embedded into Euclidean $n+1$-space, and one can take the interior of the embedding as a "simplest, canonical" choice of such a cobordant manifold. E.g. the canonical choice of manifold that has the 2-torus as boundary would be the "solid torus". Moreover, it seems to me (correct me if I'm wrong) that every other cobordant manifold can be written as the connected sum of this "canonical" cobordant manifold and a closed $n+1$-manifold.
Is there still a "canonical" choice like that for $n\geq 3$? (I'm aware that in general orientable $n$-manifolds cannot be embedded in $n+1$-space, but maybe there's some other way to "add a filling" to a $n$-manifold.)
Uh...your comment about $n < 3$ is misleading, in the sense that every orientable surface admits many embeddings in 3-space; the torus, for instance, is the boundary of a cube-with-knotted-hole, or, if you prefer, a tubular neighborhood of a trefoil knot is a solid torus whose boundary torus is also the boundary of the complement of that tubular neighborhood; after an inversion swapping some point of the tubular neighborhood with infinity, we see the torus bounding a rather different compact 3-manifold-with-boundary.
So it's a tough sell to argue that every surface bounds a "natural" solid. The cube-with-knotted-hole example is a counterexample (I believe) to the last sentence of your second paragraph.
As to whether there is some way to pick a particular cobordism, either for surfaces or for higher-dimensional null-cobordant manifolds, one that has nice properties --- I have no idea. But the surface example leaves me doubtful.