We present a time-resolved spectrophotometric study of the optical variability in the quiescent soft X-ray transient A0620-00. Superimposed on the double-humped continuum light curve are the well-known flare events, which last tens of minutes. Some of the flare events that appear in the continuum light curve are also present in the emission-line light curves. From the Balmer line flux and variations, we find that the persistent emission is optically thin. During the flare event at phase 1.15 the Balmer decrement dropped, suggesting either a significant increase in temperature or that the flares are optically thicker than the continuum. The data suggests that there are two HI emitting regions, the accretion disc and the accretion stream-disc region, with different Balmer decrements. The orbital modulation of Hα with the continuum suggests that the steeper decrement is most likely associated with the stream-disc impact region. By isolating the spectrum of the flare we find that it has a frequency power-law index of -1.40 +/- 0.20 (90 per cent confidence). The flare spectrum can also be described by an optically thin gas with a temperature in the range 10000-14000 K that covers 0.05-0.08 per cent (90 per cent confidence) of the surface of the accretion disc. Given these parameters, the possibility that the flares arise from the bright-spot cannot be ruled out. We construct Doppler images of the Hα and Hβ emission lines. Apart from showing enhanced blurred emission at the region where the stream impacts the accretion disc, the maps also show significant extended structure from the opposite side of the disc. The trailed spectra show characteristic S-wave features that can be interpreted in the context of an eccentric accretion disc.