Taking an open quantum system approach, we derive a collective equation of motion for the dynamics of a matter-wave bright soliton moving through a thermal cloud of a distinct atomic species. The reservoir interaction involves energy transfer without particle transfer between the soliton and thermal cloud, thus damping the soliton motion without altering its stability against collapse. We derive a Langevin equation for the soliton center-of-mass velocity in the form of an Ornstein-Uhlenbeck process with analytical drift and diffusion coefficients. This collective motion is confirmed by simulations of the full stochastic projected Gross-Pitaevskii equation for the matter-wave field. The system offers a pathway for experimentally observing the elusive energy-damping reservoir interaction and a clear realization of collective Brownian motion for a mesoscopic superfluid droplet.