We present a computer simulation study of a disordered two-dimensional system of localized interacting electrons in thermal equilibrium. It is shown that the configuration of occupied sites within the Coulomb gap persistently changes at temperatures much less than the gap width. This is accompanied by large time-dependent fluctuations of the site energies. The observed thermal equilibration at low temperatures suggests a possible glass transition only at T = 0. We interpret the strong fluctuations in the occupation numbers and site energies in terms of the drift of the system between multiple energy minima, which implies mobility of electrons within the Coulomb gap down to very low temperatures. Insulating properties, such as hopping conduction, appear as a result of long equilibration times associated with glassy dynamics. This may shed new light on the relation between the metal-insulator transition and glassy behavior.