We carry out an analytical and numerical study of the motion of an isolated vortex in thermal equilibrium, the vortex being defined as the point singularity of a complex scalar field $\phi(r,t)$ obeying a nonlinear stochastic Schroedinger equation. Because hydrodynamic fluctuations are included in this description, the dynamical picture of the vortex emerges as that of both a massive particle in contact with a heat bath, and as a passive scalar advected to a background random flow. We show that the vortex does not execute a simple random walk and that the probability distribution of vortex flights has non-Gaussian (exponential) tails.