Massive cosmic neutrinos change the structure formation history by suppressing perturbations on small scales. Weak lensing data from galaxy surveys probe the structure evolution and thereby can be used to constrain the total mass of the three active neutrinos. However, much of the information is at small scales where the dynamics are nonlinear. Traditional approaches with second-order statistics thus fail to fully extract the information in the lensing field. In this paper, we study constraints on the neutrino mass sum using lensing peak counts, a statistic which captures non-Gaussian information beyond the second order. We use the ray-traced weak lensing mocks from the Cosmological Massive Neutrino Simulations (MassiveNuS) and apply Large Synoptic Survey Telescope-like noise. We discuss the effects of redshift tomography, the multipole cutoff ℓmax for the power spectrum, the smoothing scale for the peak counts, and constraints from peaks of different heights. We find that combining peak counts with the traditional lensing power spectrum can improve the constraint on the neutrino mass sum, Ωm, and As by 39%, 32%, and 60%, respectively, over that from the power spectrum alone. We note that observational systematics such as baryonic effects, intrinsic alignments, and photometric redshift errors are not studied in this work, but will be an important next step.