We present spectroscopic observations of the nucleus of the Seyfert/low-ionization nuclear emission-line region galaxy NGC 1097 spanning the period 1991-1994. The goal was to monitor anticipated variations of the broad, double-peaked Balmer lines which appeared abruptly in 1991. We find that the broad Balmer lines have varied significantly over the monitoring period, both in their integrated fluxes and in their profile shapes. The integrated H-alpha flux has decreased by a factor of 2, the (H-alpha)/(H-beta) ratio has increased, and the originally asymmetric H-alpha profile has become symmetric. The decline of the H-alpha flux and the change in the (H-alpha)/(H-beta) ratio can be interpreted as consequences of either increased obscuration along the line of sight, or a decline in the ionizing continuum, but neither of these scenarios can account for the change in profile shapes. A model attributing the line emission to a precessing elliptical ring around a 10(exp 6) solar mass nuclear black hole can reproduce the observed profile variations. In this scenario, the line-emitting ring is the result of the tidal disruption of a star by the black hole. Alternative scenarios associating the broad-line emission with a collimated bipolar outflow also remain viable, but binary black holes and inhomogeneous accretion disks are disfavored by the observed pattern of variability.