A mobile-edge computing (MEC) system integrating energy harvesting (EH) techniques is a promising paradigm for supporting computation-intensive and delay-sensitive mobile applications. While computation offloading reduces users' perceived latency, EH techniques mitigate the limitation of mobile devices' battery capacity. However, when considering a scenario with multiple EH devices, the advantages of MEC systems with EH devices may be compromised due to the competition among multiple devices for available computational resources and wireless bandwidth. In this paper, the joint computation offloading and dynamic resource allocation (JCODRA) that minimizes the long-term average execution cost is formulated as a stochastic optimization problem. In particular, both the long-term average execution delay and the penalty delay are included in the optimization objective. The former intends to handle the competition and random and uncontrollable EH processes properly and the latter aims to reduce the ratio of dropped tasks. An online algorithm based on Lyapunov optimization is proposed to transform the original problem into a per-time slot deterministic problem. The results of experiments demonstrate that our algorithm significantly outperforms three representative baseline approaches.