When we think of quantum invariants, we usually think of the Jones polynomial or of the coloured HOMFLYPT. But (arguably) the simplest example of a quantum invariant of a knot or link is its Alexander polynomial. From the beginning, the central problem in the study of quantum invariants has been what do they mean topologically? The Alexander polynomial has clear algebraic topological meaning as the order of the Alexander module (first homology of the infinite cyclic cover as a module over the group of deck transformations). Can people conceptually explain (in terms of both physics and mathematics) why the representation theory of certain small quantum groups naturally gives rise to this quantity? Computationally I can understand it, but not conceptually.
A somewhat related question was already asked here.
Update: I posted on this question here and here. See also this question.
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This is far from being a full answer, but I think the key is 'how do you see the skein relation from the classical definition?'. Once you know the skein relation, it's easy to show it's a quantum invariant, and the skein relation was discovered in the 1960's, long before anyone knew about quantum groups. 


Cimasoni and Turaev have a very natural generalization of the Alexander module that's hovering around your concerns. I found out about this paper for completely different reasons  Paolo Salvatore pointed it out when we were trying to come up with an argument that the group completion of the monoid of string links can't be abelian. 

