Vocabulary/Name for Utility of a Set of Choices
What you seem to be interested is the subfield of decision theory that goes by the name of "menu choice." The starting point of this literature is the paper

Kreps, David M. "A representation theorem for preference for flexibility." Econometrica: Journal of the Econometric Society (1979): 565-577.

This paper might be directly relevant to the problem you want to address.