ABSTRACT 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 haloes 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 M500 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 per cent of BCGs are biased ∼20 per cent 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.