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We report the discovery of rapid variations of a high-velocity C iv broad absorption line trough in the quasar SDSS J141007.74+541203.3. This object was intensively observed in 2014 as a part of the Sloan Digital Sky Survey Reverberation Mapping Project, during which 32 epochs of spectroscopy were obtained with the Baryon Oscillation Spectroscopic Survey spectrograph. We observe significant (>4σ) variability in the equivalent width (EW) of the broad (Å4000 km/s wide) C iv trough on rest-frame timescales as short as 1.20 days (Å29 hr), the shortest broad absorption line variability timescale yet reported. The EW varied by Å10% on these short timescales, and by about a factor of two over the duration of the campaign. We evaluate several potential causes of the variability, concluding that the most likely cause is a rapid response to changes in the incident ionizing continuum. If the outflow is at a radius where the recombination rate is higher than the ionization rate, the timescale of variability places a lower limit on the density of the absorbing gas of n_e><≥ 3.9x10⁵/cm3. The broad absorption line variability characteristics of this quasar are consistent with those observed in previous studies of quasars, indicating that such short-term variability may in fact be common and thus can be used to learn about outflow characteristics and contributions to quasar/host-galaxy feedback scenarios.