This note is concerned with stability of monotone variational inequalities (VIs) in Hilbert spaces. Here we prove a convergence result under appropriate conditions for perturbations not only in the right hand side, but also in the convex functional and in the constraint set. We present a novel approach based on Hausdorff set convergence to handle perturbations in arbitrary closed convex constraint sets. To provide an illustrative application of our abstract stability theory we study a nonlinear nonsmooth unilateral variational problem and derive a new stability result.