A monomial (or equivariant) selfmap of a toric variety is called stable if
its action on the Picard group commutes with iteration. Generalizing work of
Favre to higher dimensions, we show that under suitable conditions, a monomial
map can be made stable by refining the underlying fan. In general, the
resulting toric variety has quotient singularities; in dimension two we give
criteria for when it can be chosen smooth, as well as examples when it cannot.