We analyze transport on a graph with multiple constraints and where the weight of the edges connecting the nodes is a dynamical variable. The network dynamics results from the interplay between a nonlinear function of the flow, dissipation, and Gaussian, additive noise. For a given set of parameters and finite noise amplitudes, the network self-organizes into one of several metastable configurations, according to a probability distribution that depends on the noise amplitude α. At a finite value α, we find a resonantlike behavior for which one network topology is the most probable stationary state. This specific topology maximizes the robustness and transport efficiency, it is reached with the maximal convergence rate, and it is not found by the noiseless dynamics. We argue that this behavior is a manifestation of noise-induced resonances in network self-organization. Our findings show that stochastic dynamics can boost transport on a nonlinear network and, further, suggest a change of paradigm about the role of noise in optimization algorithms.