We investigate the constructive role of an external noise signal, in the form of a low-rate Poisson sequence of pulses supplied to all inputs of a spiking neural network, consisting in maintaining for a long time or even recovering a memory trace (engram) of the image without its direct renewal (or rewriting). In particular, this unique dynamic property is demonstrated in a single-layer spiking neural network consisting of simple integrate-and-fire neurons and memristive synaptic weights. This is carried out by preserving and even fine-tuning the conductance values of memristors in terms of dynamic plasticity, specifically spike-timing-dependent plasticity-type, driven by overlapping pre- and postsynaptic voltage spikes. It has been shown that the weights can be to a certain extent unreliable, due to such characteristics as the limited retention time of resistive state or the variation of switching voltages. Such a noise-assisted persistence of memory, on one hand, could be a prototypical mechanism in a biological nervous system and, on the other hand, brings one step closer to the possibility of building reliable spiking neural networks composed of unreliable analog elements.