Many of the scientific applications for X-ray free-electron lasers seek to exploit the ultrashort pulse durations of intense X-rays to obtain femtosecond time resolution of various processes in a "pump-probe" scheme. One of the limiting factors for such experiments is the timing jitter between the X-rays and ultrashort pulses from more conventional lasers operating at near-optical wavelengths. In this work, we investigate the potential of using X-ray-induced changes in the optical second harmonic generation efficiency of a nonlinear crystal to retrieve single-shot arrival times of X-ray pulses with respect to optical laser pulses. Our experimental results and simulations show changes to the efficiency of the second harmonic generation of 12%, approximately three times larger than the measured changes in the transmission of the 800 nm center-wavelength fundamental pulse. Further experiments showing even larger changes in the transmission of 400 nm center-wavelength pulses show that the mechanism of the second harmonic generation efficiency modulation is mainly the result of X-ray-induced changes in the linear absorption coefficients near 400 nm. We demonstrate and characterize a cross-correlation tool based on this effect in reference to a previously demonstrated method of X-ray/optical cross-correlation.