Noise is a pervasive aspect that impacts various systems and environments, from mobile radio channels to biological systems. Within the framework of complex networks, noise poses significant challenges for functionality and performance. In this paper, we investigate the dynamics of a well-known type of locally-coupled computing networks, Memristor Cellular Nonlinear Networks (M-CNNs), in the presence of noise at their interconnection weights, introducing the concept of stochastic weights. In particular, we analyze the effect of noise originating from the synaptic memristors by incorporating both deterministic and stochastic components into synaptic weights, investigating how device-to-device variability and noise affect network performance. Based on the well-established theory of CNNs, we are extending the stability criteria to incorporate synaptic memristor non-idealities and we provide a theoretical framework to analyze their effect on system's performance. In this work, we employ the physics-based Jülich Aachen Resistive Switching Tools (JART) model to study Valence Change Memory (VCM) devices as synapses within our theoretical framework. We investigate the impact of device variability and noise, utilizing statistical properties derived from experimental data reported in the literature. We demonstrate the efficacy of noisy M-CNNs in performing the edge detection task, an example of fundamental image processing applications.