> a proper (i.e. non-stupid) generalization of the Conway polynomial to some multi-variable 
> polynomial for links seems to be something people are after

How about this: http://front.math.ucdavis.edu/0312.5007

> What is the Conway polynomial supposed to be measuring?

Cochran, Levine and others gave various good answers - see references in the linked paper.
Thinking of what the *multi-variable* Conway polynomial is measuring seems to give a much clearer picture, at least in the 2-variable case. What is less clear is *why* it does so -
I mean that there should be more conceptual proofs.