Utilizing the intelligence at the network edge, edge computing paradigm emerges to provide time-sensitive computing services for Internet of Things. In this paper, we investigate sustainable computation offloading in an edge-computing system that consists of energy harvesting-enabled mobile devices (MDs) and a dispatcher. The dispatcher collects computation tasks generated by IoT devices with limited computation power, and offloads them to resourceful MDs in exchange for rewards. We propose an online Rewards-optimal Auction (RoA) to optimize the long-term sum-of-rewards for processing offloaded tasks, meanwhile adapting to the highly dynamic energy harvesting (EH) process and computation task arrivals. RoA is designed based on Lyapunov optimization and Vickrey-Clarke-Groves auction, the operation of which does not require a prior knowledge of the energy harvesting, task arrivals, or wireless channel statistics. Our analytical results confirm the optimality of tasks assignment. Furthermore, simulation results validate the analytical analysis, and verify the efficacy of the proposed RoA.