In this paper, a new Cellular Automata (CA) Model, named Cellular Automata with Random Memory, has been introduced. The new model is in fact, constructed by randomizing the choosing memory operation of a Cellular Automata with memory. Therefore, the model has a potential usage for application like Random Number Generators, Physical Unclonable Functions. The introduced model and other CA models are represented in an abstract form. Delay lines comprising sequentially connected logic gates are introduced in the implementation of the memory part of the introduced model. Due to the process of variations and jitter effect, the delay line becomes beneficial in the implementation of the proposed Cellular Automata with Random Memory. The introduced model is implemented on a FPGA using delay lines and also using other alternative methods.