We report the first comprehensive study of emission-line variability in an accreting young brown dwarf. We have collected 14 high-resolution optical spectra of 2MASSW J1207334-393254 (M8), a likely member of the nearby 8 million year old TW Hydrae association with a recently identified planetary mass companion, in three observing runs between 2005 January-March on the Magellan Clay telescope. These spectra show a variety of emission lines that are commonly seen in classical T Tauri stars. The Hα line, in particular, shows dramatic changes in shape and intensity in our data set, on timescales of both several weeks and several hours. In spectra from late January, the line is relatively weak and only slightly asymmetric. Spectra from mid and late March show intense, broad (10% width ~280 km s^-1), and asymmetric Hα emission, indicative of ongoing disk accretion. Based on empirical diagnostics, we estimate that the accretion rate could have changed by a factor of 5-10 over ~6 weeks in this brown dwarf, which may be in the final stages of accreting from its disk. March spectra also reveal significant "quasi-periodic" changes in the Hα line profile over the course of a night, from clearly double-peaked to nearly symmetric. These nightly profile changes, roughly consistent with the brown dwarf's rotation period, could be the result of a redshifted absorption feature coming into and out of our line of sight; when the profile is double-peaked we may be looking into an accretion column, flowing from the inner disk edge-on to the central object, indicating that the accretion is probably channeled along the magnetic field lines. Our findings provide strong support for the magnetospheric accretion scenario, and thus for the existence of large-scale magnetic fields, in the substellar regime. We also present the first high-resolution optical spectrum of SSSPM J1102-3431 (M8.5), which has recently been identified as another likely substellar member of the TW Hydrae association. Its emission lines are relatively narrow and fairly symmetric, suggesting that it is accreting only very weakly, if at all.