  The technique of weak-lensing aperture mass densitometry, so called the
zeta-statistic, has recently been popular in actual observations for
measurement of individual cluster mass. It has however been anticipated that
the line-of-sight projection by foreground and background matter can adversely
affect the cluster mass determination with not only substantial error
dispersion but also a sizable positive systematic bias. Additionally, the
finite number of background galaxies even at a reasonable observing depth can
also introduce Poisson noise to the mass estimate. In this paper, we
quantitatively investigate the degree of errors separately contributed by the
two sources to the mass determination of those galaxy clusters with
M_{200}>10^{14}M_{\odot}. We find that the aperture mass of zeta-statistic
turns out to be a mass estimator of much reduced systematic bias, due to the
cancellation by the positively biased local background mass sheet. However, the
error dispersion of M_{200} arising from both projection effect and Poisson
noise is found to be still sizable (40%-90%), even for the shear-selected,
clean sample where multiple clusters located within a suitable projected
aperture are removed. We also investigate how to remedy this large-error
problem in weak lensing measurements, and propose a plausible alternative mass
estimator, M(<theta_{1000}), an aperture mass measured within about half the
virial radius. The aperture mass M(<theta_{1000}) is free of bias and has a
substantially reduced error dispersion, 39% for the worst case of high-z,
low-mass clusters, that can be smaller than the error dispersion of M_{200} as
much as a factor 3.
