We study the effect of noise on the switching dynamics of memristors in a phenomenological three-stable model of conductive filament (CF) formation. The approach is based on the assumption that in a memristive system the formation of CF undergoes nucleation stages similar to the thermodynamic phase transition model.  The nucleation evolves in two steps through an intermediate metastable state towards a long-lived state. Each state corresponds to one of the local minima of the free energy profile. To describe the switching dynamics, under the action of external voltage and in the presence of the noise, we use the mathematical apparatus of the Fokker-Planck equation. We investigate the relaxation times of the system and the transition times between the states of the multistable system. We find nonmonotonic behaviours of the relaxation time as a function of the noise intensity, which are signatures of two noise-induced phenomena. Specifically, we observe the noise enhanced stability for the intermediate metastable state and noise delayed decay for the unstable nonequilibrium state. Moreover, we observe the stochastic resonance phenomenon in the system investigated. The main relevant conclusion is that the external noise can improve the switching dynamics of memristors.