  Weak lensing measurements of cluster masses are necessary for calibrating
mass-observable relations (MORs) to investigate the growth of structure and the
properties of dark energy. However, the measured cluster shear signal varies at
fixed mass M_200m due to inherent ellipticity of background galaxies,
intervening structures along the line of sight, and variations in the cluster
structure due to scatter in concentrations, asphericity and substructure. We
use N-body simulated halos to derive and evaluate a weak lensing circular
aperture mass measurement M_ap that minimizes the mass estimate variance <(M_ap
- M_200m)^2> in the presence of all these forms of variability. Depending on
halo mass and observational conditions, the resulting mass estimator improves
on M_ap filters optimized for circular NFW-profile clusters in the presence of
uncorrelated large scale structure (LSS) about as much as the latter improve on
an estimator that only minimizes the influence of shape noise. Optimizing for
uncorrelated LSS while ignoring the variation of internal cluster structure
puts too much weight on the profile near the cores of halos, and under some
circumstances can even be worse than not accounting for LSS at all. We briefly
discuss the impact of variability in cluster structure and correlated
structures on the design and performance of weak lensing surveys intended to
calibrate cluster MORs.
