The fluctuation-dissipation theorem for the static dielectric response function of systems of ions with inducible point dipoles is derived. It is shown that the static longitudinal dielectric function is determined by spatial correlations of both charge and dipole-moment density fluctuations. Moreover, it is deduced that the long-wavelength behavior of the charge structure factor for polarizable ion systems is different from that for systems of rigid ions. Molecular dynamics simulation results of rigid and polarizable ion models for molten AgI and NaI are reported.