  Weak-lensing measurements of the masses of galaxy clusters are commonly based
on the assumption of spherically symmetric density profiles. Yet, the cold dark
matter model predicts the shapes of dark matter halos to be triaxial. Halo
triaxiality, and the orientation of the major axis with respect to the line of
sight, are expected to be the leading cause of intrinsic scatter in
weak-lensing mass measurements. The shape of central cluster galaxies
(Brightest Cluster Galaxies; BCGs) is expected to follow the shape of the dark
matter halo. Here we investigate the use of BCG ellipticity as predictor of the
weak-lensing mass bias in individual clusters compared to the mean. Using weak
lensing masses $M^{\rm WL}_{500}$ from the Weighing the Giants project, and
$M_{500}$ derived from gas masses as low-scatter mass proxy, we find that, on
average, the lensing masses of clusters with the roundest / most elliptical 25%
of BCGs are biased $\sim 20$% high / low compared to the average, as
qualitatively predicted by the cold dark matter model. For cluster cosmology
projects utilizing weak-lensing mass estimates, the shape of the BCG can thus
contribute useful information on the effect of orientation bias in weak lensing
mass estimates as well as on cluster selection bias.
