The concept of discrete memristors has recently attracted significant interest. Existing studies show that integrating them into classical chaotic maps can enhance the underlying chaotic dynamics. However, fractional discrete memristors and their coupled maps have received comparatively little attention. Recent findings suggest that introducing fractional discrete memristors into different discrete systems can further increase the complexity of chaotic behavior of these systems, the broad potential of fractional discrete memristors in applications such as cellular neural networks, modulators, sensors, and chaotic systems. Motivated by this, the present work investigates the mathematical model of fractional discrete memristors and their role in constructing new discrete maps by coupling them with well-known discrete systems. We first introduce the basic notion of discrete memristors along with several classical models. Then, by embedding the ideal discrete memristor into the Caputo-like difference operator, a fractional discrete memristor is formulated. The resulting device exhibits characteristic pinched hysteresis loops under bipolar periodic excitation, thereby satisfying the defining properties of a memristor. As an application, we propose a fractional discrete memristor Hénon map. The findings confirm that fractional discrete memristors significantly enrich the complexity of chaotic dynamics.