We study the motion of the hypersurface $(γ_t)_{t\geq 0}$ evolving according to the mean curvature perturbed by $\dot{w}^Q$, the formal time derivative of the $Q$-Wiener process ${w}^Q$, in a two dimensional bounded domain. Namely, we consider the equation describing the evolution of $γ_t$ as a stochastic partial differential equation (SPDE) with a multiplicative noise in the Stratonovich sense, whose inward velocity $V$ is determined by $V=κ\,+\,G \circ \dot{w}^Q$, where $κ$ is the mean curvature and $G$ is a function determined from $γ_t$. Already known results in which the noise depends on only time variable is not applicable to our equation. To construct a local solution of the equation describing $γ_t$, we will derive a certain second order quasilinear SPDE with respect to the signed distance function determined from $γ_0$. Then we construct the local solution making use of probabilistic tools and the classical Banach fixed-point theorem on suitable Sobolev spaces.