Using a primitive model approach, we analyze the influence of ion specific adsorption on the phase behavior of charged lamellar systems. The presence of a weak short-ranged surface potential, attracting monovalent counterions, induces a phase separation, where the separate phases have different repeat distance. If the adsorption potential is very weak, the more narrow phase never forms. An opposite behavior is found for strong surface affinities. Both Monte Carlo simulations and a recently developed correlation-corrected Poisson-Boltzmann theory are adopted, with a nearly quantitative agreement between the approaches. Different counterions are discriminated by the adsorption potential strength, and with physically reasonable values, experimental observations on these systems are well reproduced. The study highlights the importance of electrostatic correlations, even though only monovalent ions are present.