Memristors are emerging as a potential candidate for next-generation memory technologies, promising to deliver non-volatility at performance and density targets which were previously the domain of SRAM and DRAM. Silicon Physically Unclonable Functions (PUFs) have been introduced as a relatively new security primitive which exploit manufacturing variation resulting from the IC fabrication process to uniquely fingerprint a device instance or generate device-specific cryptographic key material. While silicon PUFs have been proposed which build on traditional memory structures, in particular SRAM, in this paper we present a memristor-based PUF which utilizes a weak-write mechanism to obtain cell behaviour which is influenced by process variation and hence usable as a PUF response. Using a model-based approach we evaluate memristor PUFs under random process variations and present results on the performance of this new PUF variant.