We introduce a general framework for the study of numerical approximations of a certain class of solutions, called stable solutions, of second order mean-field game systems for which uniqueness of solutions is not guaranteed. To illustrate the approach, we focus on a very simple example of stationary second-order MFG system with local coupling and a quadratic Hamiltonian. We provide sufficient conditions for the stability of solutions and it turns out that stability is a generic property of the MFG. We then re-express the solutions of the system as zeros of a well chosen nonlinear map and establish the fact that stable solutions are regular points of this map. This fact is then used to study the approximation of solutions by finite elements and the local convergence of Newton's method in infinite dimension.