  We study the stochastic mass-conserving Allen-Cahn equation posed on a
bounded two-dimensional domain with additive spatially smooth space-time noise.
This equation associated with a small positive parameter describes the
stochastic motion of a small almost semicircular droplet attached to domain's
boundary and moving towards a point of locally maximum curvature. We apply
It\^o calculus to derive the stochastic dynamics of the droplet by utilizing
the approximately invariant manifold introduced by Alikakos, Chen and Fusco for
the deterministic problem. In the stochastic case depending on the scaling, the
motion is driven by the change in the curvature of the boundary and the
stochastic forcing. Moreover, under the assumption of a sufficiently small
noise strength, we establish stochastic stability of a neighborhood of the
manifold of droplets in $L^2$ and $H^1$, which means that with overwhelming
probability the solution stays close to the manifold for very long time-scales.
