This paper is concerned with robust stabilization state feedback synthesis for discrete-time linear systems characterized by random polytopes. In particular, we develop what we call an S-variable approach to the synthesis. S-variables are also called auxiliary variables in some references, and known to be useful for reducing conservativeness in robustness analysis and synthesis. In this paper, we show a sort of stochastic version of the conventional (i.e., deterministic) robust stability condition with S-variables, and numerically illustrate that introducing S-variables is also effective in the synthesis for systems characterized by random polytopes.