This question stems from Dick Lipton's recent blog post on the Axiom of Choice. I asked there but got no takers. I promise I'm not an inept Googler, but I couldn't find a satisfactory answer. I suspect universal in this context means computable by a universal Turing machine, or something close to that, but I'd like to know for sure.
In that argument, he just means that g is defined on all the two-element subsets that may arise in the argument, that is, for a given family F of four-element sets, g(A) should be defined on any two-element set A that is a subset of a four-element set in F.
The reason he needs to assume that is that he cannot allow that we need to choose the choice function g itself to be used with each separate A. There are many choice functions that work for families of 2-element sets, and different choices of g will give rise to different functions on the families of four-element sets. The way the argument works is that you make one choice of the function g that works on all the two element sets that arise, and then you define the choice function on the given family of four-element sets by the clever construction in the article.
In particular, he is not using some technical meaning of universal.