The Kerr effect measures a change in the refractive index proportional to the intensity of the applied electric field, and its ultrafast implementation has been widely used to investigate the nonlinear optical properties of many different systems. More recently, the same mechanism has been exploited with terahertz light pulses to access resonant processes involving lattice vibrations or broken-symmetry collective modes. Here we provide experimental evidence and theoretical description that in insulating SrTiO3 the terahertz Kerr effect admits a sizeable response due to lattice degrees of freedom. This ionic contribution exploits the ability of strong terahertz pulses to excite multiple infrared phonons at an intermediate step before or after the usual off-resonant electronic excitations responsible for the electronic Kerr effect. The mechanism is identified thanks to a quantitative theoretical model of the time and polarization dependence of both the electronic and phononic responses. Such a ionic Kerr effect provides a tunable mechanism to modulate the refractive index on ultrashort time-scales and, given the capability of terahertz-driven phonons to couple to order parameters and to drive materials towards metastable states which may not be accessible at thermal equilibrium, it can be used to investigate the electron-phonon coupling across various phase transitions.