  We explore the effects of event-by-event fluctuations of Hanbury Brown-Twiss
(HBT) radii and show how they can be observed experimentally. The relation of
measured HBT radii extracted from ensemble-averaged correlation functions to
the mean of their event-by-event probability distribution is clarified. We
propose a method to experimentally determine the mean and variance of this
distribution and test it on an ensemble of fluctuating events generated with
the viscous hydrodynamic code VISH2+1. Using the same code, the sensitivity of
the mean and variance of the HBT radii to the specific QGP shear viscosity
$\eta/s$ is studied. We report sensitivity of the mean pion HBT radii and their
variances to the temperature dependence of $\eta/s$ near the quark-hadron
transition at a level similar (10-20%) to that which was previously observed
for elliptic and quadrangular flow of charged hadrons [1].
