Artificial Intelligence (AI) at the edge has become a hot subject of the recent technology-minded publications. The challenges related to IoT nodes gave rise to research on efficient hardware-based accelerators. In this context, analog memristor devices are crucial elements to efficiently perform the multiply-and-add (MAD) operations found in many AI algorithms. This is due to the ability of memristor devices to perform in-memory-computing (IMC) in a way that mimics the synapses in human brain. Here, we present a novel planar analog memristor, namely NeuroMem, that includes a partially reduced Graphene Oxide (prGO) thin film. The analog and non-volatile resistance switching of NeuroMem enable tuning it to any value within the RON and ROFF range. These two features make NeuroMem a potential candidate for emerging IMC applications such as inference engine for AI systems. Moreover, the prGO thin film of the memristor is patterned on a flexible substrate of Cyclic Olefin Copolymer (COC) using standard microfabrication techniques. This provides new opportunities for simple, flexible, and cost-effective fabrication of solution-based Graphene-based memristors. In addition to providing detailed electrical characterization of the device, a crossbar of the technology has been fabricated to demonstrate its ability to implement IMC for MAD operations targeting fully connected layer of Artificial Neural Network. This work is the first to report on the great potential of this technology for AI inference application especially for edge devices.