Hanbury-Brown-Twiss interferometry is a technique which yields effective widths (i.e., "HBT radii") of homogeneity regions in the fireballs produced in heavy ion collisions. Because the initial conditions of these collisions are stochastically fluctuating, the measured HBT radii also exhibit variation on an event-by-event basis. However, HBT measurements have, to date, been performed only on an ensemble-averaged basis, due to inherent limitations of finite particle statistics. In this paper, we show that experimental measurements to date are best characterized theoretically as weighted averages of the event-by-event HBT radii, and we propose a new method for extracting experimentally both the arithmetic mean and the variance of the event-by-event distribution of HBT radii. We demonstrate the extraction of the mean and variance of this distribution for a particular ensemble of numerically generated events, and offer some ideas to extend and generalize the method to enable measurement of higher moments of the HBT distribution as well.