A general superconvergence result is established for the stabilized finite element approximations for the stationary Stokes equations. The superconvergence is obtained by applying the $L^2$ projection method for the finite element approximations and/or their close relatives. For the standard Galerkin method, existing results show that superconvergence is possible by projecting directly the finite element approximations onto properly defined finite element spaces associated with a mesh with different scales. But for the stabilized finite element method, the authors had to apply the $L^2$ projection on a trivially modified version of the finite element solution. This papers shows how the modification should be made and why the $L^2$ projection on the modified solution has superconvergence. Although the method is demonstrated for one class of stabilized finite element methods, it can certainly be extended to other type of stabilized schemes without any difficulty. Like other results in the family of $L^2$ projection methods, the superconvergence presented in this paper is based on some regularity assumption for the Stokes problem and is valid for general stabilized finite element method with regular but non-uniform partitions.