Electronic bound states around charged impurities in two-dimensional systems with structural inversion asymmetry can be described in terms of a two-dimensional hydrogen atom in the presence of a Rashba spin-orbit interaction. Here, the energy levels of the bound electron are evaluated numerically as a function of the spin-orbit interaction, and analytic expressions for the weak and strong spin-orbit coupling limits are compared with the numerical results. It is found that, besides the level splitting due to the lack of inversion symmetry, the energy levels are lowered for sufficiently strong spin-orbit coupling, indicating that the electron gets more tightly bound to the ion as the spin-orbit interaction increases. Similarities and differences with respect to the two-dimensional Frohlich polaron with the Rashba coupling are discussed.