Mobile edge computing (MEC), as a new distributed computing model, satisfies the low energy consumption and low latency requirements of computation-intensive services. The task offloading of MEC has become an important research hotspot, as it solves the problems of insufficient computing capability and battery capacity of Internet of things (IoT) devices. This study investigates task offloading scheduling in a dynamic MEC system. By integrating energy harvesting technology into IoT devices, we propose a hybrid energy supply model. We jointly optimize local computing, offloading duration, and edge computing decisions to minimize system cost. On the basis of stochastic optimization theory, we design an online dynamic task offloading algorithm for MEC with a hybrid energy supply called DTOME. DTOME can make task offloading decisions by weighing system cost and queue stability. We quote dynamic programming theory to obtain the optimal task offloading strategy. Simulation results verify the effectiveness of DTOME, and show that DTOME entails lower system cost than two baseline task offloading strategies.