In the modern language, sets with all subset sums pairwise distinct are often called dissociated. Maximal dissociated subsets of a given set are important in Additive Combinatorics and Fourier analysis, although I cannot recall anybody ever addressing the algorithmic aspect.
Yet another reference you may find useful: http://math.haifa.ac.il/~seva/Papers/DisBases.pdf.
Added: taking into account the interpretation of a maximal dissociated subset of a given set $A$ as a "basis of $A$ over the set ${-1,0,1}$", it may be reasonable to try and adjust one of the standard algorithms of finding a maximal independent subset of a given set in a vector space.
