This study explores the nonlinear dynamics of a fractional-order discrete FitzHugh–Nagumo (FHN) neuron model with memristive autapse, formulated as a fractional generalization of the classical integer-order counterpart. The fractional formulation is constructed using Caputo-like discrete fractional operators, thereby incorporating memory effects into the system. Through extensive numerical analysis, we examine the dynamical properties of the proposed model under varying fractional orders. The investigation employs bifurcation diagrams, phase portraits, and the maximum Lyapunov exponent to characterize transitions between periodicity and chaos. Furthermore, advanced complexity measures, including the 0–1 test for chaos, SE complexity, and sample entropy, are applied to provide a broader perspective of the system’s irregular behaviors. The results reveal that the inclusion of fractional orders enriches the dynamical landscape, leading to multistability, coexisting attractors, and enhanced sensitivity to initial conditions.