Probably the answer depends on the context, where you need the generalized concept.
Unfortunately, usually the linearized operator is not bounded anymore, because it contains differential operators.
As pointed out by Mikael, the Lipschitz constant may be one possibility. For example, in the Crandall-Liggett theory of nonlinear semigroups the Hille-Yosida generation theorem on linear contraction semigroups is generalized to nonlinear contractions. Here, definitely the Lipschitz constant replaces the role of the operator norm. Or in the Banach fixed point theorem the convergence of the geometric series is generalized to iterations of nonlinear maps.
But there might be other answers, I am really curious.