On-device sequential recommendation (SR) systems are designed to make local inferences using real-time features, thereby alleviating the communication burden on server-based recommenders when handling concurrent requests from millions of users. However, the resource constraints of edge devices, including limited memory and computational capacity, pose significant challenges to deploying efficient SR models. Inspired by the energy-efficient and sparse computing properties of deep Spiking Neural Networks (SNNs), we propose a cost-effective on-device SR model named SSR, which encodes dense embedding representations into sparse spike-wise representations and integrates novel spiking filter modules to extract temporal patterns and critical features from item sequences, optimizing computational and memory efficiency without sacrificing recommendation accuracy. Extensive experiments on real-world datasets demonstrate the superiority of SSR. Compared to other SR baselines, SSR achieves comparable recommendation performance while reducing energy consumption by an average of 59.43%. In addition, SSR significantly lowers memory usage, making it particularly well-suited for deployment on resource-constrained edge devices.