We study the effects of thermally induced capillary waves in the fragmentation of a liquid ligament into multiple nanodroplets. Our numerical implementation is based on a fluctuating lattice Boltzmann (LB) model for nonideal multicomponent fluids, including nonequilibrium stochastic fluxes mimicking the effects of molecular forces at the nanoscales. We quantitatively analyze the statistical distribution of the breakup times and the droplet volumes after the fragmentation process at changing the two relevant length scales of the problem, i.e., the thermal length scale and the ligament size. The robustness of the observed findings is also corroborated by quantitative comparisons with the predictions of sharp interface hydrodynamics. Beyond the practical importance of our findings for nanofluidic engineering devices, our study also explores a novel application of LB in the realm of nanofluidic phenomena.