The two-dimensional layered material CuInP2S6 (CIPS) has attracted significant research attention due to its nontrivial physical properties, including room-temperature ferroelectricity at the ultrathin limit and substantial ionic conductivity. Despite many efforts to control its ionic conductance and develop electronic devices, such as memristors, improving the stability of these devices remains a challenge. This work presents a highly stable threshold-switching device based on the Cu/CIPS/graphene heterostructure, achieved after a comprehensive investigation of the activation of Cu's ionic conductivity. The device exhibits exceptional threshold-switching performance, including good cycling endurance, a high on/off ratio of up to 104, low operation voltages, and an ultrasmall subthreshold swing of less than 1.8 mV/decade for the resistance-switching process. Through temperature-dependent electrical and Raman spectroscopy measurements, the stable resistive-switching mechanism is interpreted with a drifting and diffusion model of Cu ions under the electric field, rather than the conventional conducting filament mechanism. These results make the layered ferroionic CIPS material a promising candidate for information storage devices, demonstrating a compelling approach to achieving high-performance threshold-switching memristor devices.