Recently, mobile edge computing (MEC) technology was developed to mitigate the overload problem in networks and cloud systems. An MEC system computes the offloading computation tasks from resource-constrained Internet of Things (IoT) devices. In addition, several convergence technologies with renewable energy resources (RERs) such as photovoltaics have been proposed to improve the survivability of IoT systems. This paper proposes an MEC integrated with RER system, which is referred to as energy-harvesting (EH) MEC. Since the energy supply of RERs is unstable due to various reasons, EH MEC needs to consider the state-of-charge (SoC) of the battery to ensure system stability. Therefore, in this paper, we propose an offloading scheduling algorithm considering the battery of EH MEC as well as the service quality of experience (QoE). The proposed scheduling algorithm consists of a two-stage operation, where the first stage consists of admission control of the offloading requests and the second stage consists of computation frequency scheduling of the MEC server. For the first stage, a non-convex optimization problem is designed considering the computation capability, SoC, and request deadline. To solve the non-convex problem, a greedy algorithm is proposed to obtain approximate optimal solutions. In the second stage, based on Lyapunov optimization, a low-complexity algorithm is proposed, which considers both the workload queue and battery stability. In addition, performance evaluations of the proposed algorithm were conducted via simulation. However, this paper has a limitation in terms of verifying in a real-world scenario.