Does this problem related to subset sum have a name?
I've never heard of this problem before, but it's clearly NP-hard.

There's a simple polynomial time reduction from subset-sum to this problem.  All you have to do is take an instance of subset-sum such as:  {x1, x2, x3, ... , x_n} ...and map it to this instance of the problem you describe:  S1 = {x1, x2, x3, ..., x_n}; S2 = {0}.

(I initially misread your question and thought that you were talking about a decision problem; since that was not what you meant, I cannot say anything beyond whether or not it is NP-hard except that it is EXPTIME.  It is likely not NP; see the comments below.)

Either way, I don't think you are going to find any good solution or approximation algorithms.