Mobile edge computing (MEC)-enabled Internet of Things (IoT) networks have been deemed a promising paradigm to support massive energy-constrained and computation-limited IoT devices. IoT with mobility has found tremendous new services in the 5G era and the forthcoming 6G eras such as autonomous driving and vehicular communications. However, mobility of IoT devices has not been studied in the sufficient level in the existing works. In this paper, the offloading decision and resource allocation problem is studied with mobility consideration. The long-term average sum service cost of all the mobile IoT devices (MIDs) is minimized by jointly optimizing the CPU-cycle frequencies, the transmit power, and the user association vector of MIDs. An online mobility-aware offloading and resource allocation (OMORA) algorithm is proposed based on Lyapunov optimization and Semi-Definite Programming (SDP). Simulation results demonstrate that our proposed scheme can balance the system service cost and the delay performance, and outperforms other offloading benchmark methods in terms of the system service cost.