Observations of the ultraviolet continuum of the X-ray binary system Her X-1/HZ Herculis with the Space Telescope Imaging Spectrograph on the Hubble Space Telescope show quasiperiodic oscillations (QPOs) at frequencies of 8+/-2 and 43+/-2 mHz, with rms amplitudes of 2 and 4% of the steady flux. Observations with the Keck telescope confirm the presence of the higher frequency QPO in the optical continuum, with a rms amplitude of 1.6+/-0.2%. The QPOs are most prominent in the HST data near phi=0.5 (where phi=0 is the middle of the X-ray eclipse), suggesting that they arise not in the accretion disk but on the X-ray heated face of the companion star. We discuss scenarios in which the companion star reprocesses oscillations in the disk which are caused by either Keplerian rotation or a beat frequency between the neutron star spin and Keplerian rotation at some radius in the accretion disk.