Neutron noise in nuclear power reactors refers to the small fluctuations around the average neutron flux at steady state resulting from time-dependent perturbations inside the core. The neutron noise equations in the frequency domain can be solved using Monte Carlo simulation codes, which are capable of obtaining reference solutions involving almost no approximations but are hindered by severe issues affecting the statistical convergence: The simultaneous presence of positive and negative particles, which is required by the nature of the complex noise equations, leads to catastrophically large variance in the tallies. In this work, we consider the important case of neutron noise problems induced by mechanical vibrations. First, we derive a new exact sampling strategy for the noise source. Then, building upon our previous findings in other contexts, we show that weight cancellation methods can be highly beneficial in dealing with the presence of negative weights, enabling extremely large gains in the figure of merit. We successfully demonstrate our results on a benchmark configuration consisting of a fuel assembly with a vibrating pin, and we discuss possible pathways for further improvements.