We present a spectrophotometric study of short-term optical variability in the quiescent black hole X-ray transient V404 Cyg. This includes two nights of high-time-resolution Hα spectroscopy with which we resolve much of the time variability, and a further six nights of archival spectroscopy with lower time resolution but higher spectral resolution. We find significant variability in most of the data considered, with both the Hα line and the continuum often varying in a correlated way. This includes both dramatic flares lasting a few hours in which the line flux nearly doubles and lower-level flickering. The strongest flares involve development of asymmetry in the line profile, with the red wing usually strongest independent of orbital phase. It is unclear why this is the case, but we discuss several possible explanations. We consider the energetics of the flares and compare with plausible models including chromospheric activity on the companion star, local magnetic reconnection events within the disc and varying irradiation from close to the black hole. Based on the line profile changes during the flares, we conclude that the most likely origin for the variability is variable photoionization by the central source, although local flares within the disc cannot be ruled out.