We present a spectrophotometric study of short-term optical variability in the quiescent black hole X-ray transient V404 Cyg focusing on two nights of Hα spectroscopy. We find significant variability, with both the Hα line and the continuum 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. Based on the line profile changes during the flares, we conclude that the most likely origin for the variability is variable photoionisation by the central source.