Mixture-of-Experts (MoE) models offer a promising path for scaling model capacity, yet their massive memory footprint poses significant challenges for deployment on resource-constrained edge devices. Existing solutions, such as static pruning or dynamic offloading, often struggle to balance model accuracy with inference latency due to irreversible information loss or prohibitive I/O overhead. In this paper, we propose LightMoE, a novel framework for memory-efficient MoE inference that exploits the inherent functional redundancy and temporal locality of expert activation. LightMoE employs a frequency-aware expert initialization strategy to retain a compact core of resident experts and introduces a similarity-based redirection mechanism to compensate for missing experts without incurring I/O costs. Furthermore, it incorporates a lightweight runtime manager that performs coarse-grained, task-level expert replacement to adapt to shifting data distributions. Empirical evaluations on representative edge platforms demonstrate that LightMoE achieves a superior accuracy-efficiency trade-off, improving average accuracy by 4.3% over static pruning and 2.4% over dynamic swapping methods, while maintaining inference latency comparable to strictly pruned models.