The question is the following: suppose I have manifolds $N$ and $M$ both with boundary, and I have a covering map $\phi$ from $\partial N$ to $\partial M.$ The question is: when is there a covering map $\tilde{\phi}: N \rightarrow M$ which restricts to $\phi?$ When $N, M$ are surfaces, the covering map of the boundaries is given by its monodromy, and there is a $\mathbb{Z}/2 \mathbb{Z}$ obstruction (the sign of the monodromy permutation should be $+1.$ (put a different way: each component of the boundary gets mapped to a collection of cycles. The covering can be extended if and only if the sum of the numbers of even cycles over all boundary components is even). This was proved by Husemoller and Gleason in Husemoller's thesis in the late fifties. I know of no general result in higher dimensions (even in dimension three), from which I conjecture that the question is known to be hard.

There's an algorithm to tell if such a map exists if $N$ and $M$ are compact 3manifolds. Given a covering map $\partial N\to \partial M$ of degree $n$, any extension to a covering $N\to M$ must be a covering of the same degree $n$. Construct all covers of $M$ of degree $n$ (this amounts to finding all reps. of $\pi_1 M$ to the symmetric group $S_n$), and determine for which of these covers there is a lift of the maps $\partial N\to \partial M$. This reduces the question to whether given a homeomorphism $\partial N \to \partial M$ extends to a homeomorphism $N\to M$ by replacing $M$ with the $n$fold covers together with the lifts of the boundary maps (in fact this part of the argument works in any dimensions). So we will now restrict to the case that the map $\partial N\to \partial M$ is a homeomorphism. I'll now assume that $N$ and $M$ are 3dimensional with nontrivial boundary (otherwise you're just asking for the homeomorphism problem), and irreducible (one may reduce to the irreducible case in the usual fashion using the KneserMilnor decomposition). Then $N$ and $M$ are Haken. Put a boundary pattern in $\partial N$ which has no nontrivial automorphisms, and use the homeomorphism to $\partial M$ to transfer the boundary pattern to $\partial M$. By Theorem 6.1.6 of Matveev's book, there is an algorithm to tell if there is a homeomorphism of $N$ with $M$ which preserves the boundary pattern. This algorithm will then tell if the homeomorphism $\partial N\to \partial M$ extends to a homeomorphism $N\to M$ since the boundary pattern has no automorphisms. I don't expect this homemorphism extension problem to be solvable in higher dimensions, unless possibly one restricts to some very special class of manifolds. 

