The deployment of unmanned aerial vehicles (UAVs) in wireless communication systems promises to provide services for devices with limited or without infrastructure coverage. With the emergence of diverse Internet of Things (IoT) applications, there is an ever-increasing demand for computation resources of IoT mobile devices (IMDs) in UAV communication scenarios. Motivated by this, we consider a UAV-aided edge computing scenario and study the task offloading problem between the IMDs and the UAV, aiming to minimize the overall energy consumption for accomplishing the tasks. An alternative optimization algorithm is proposed as our solution, which jointly optimizes task offloading decision-making, bit allocation during transmission, and the UAV trajectory. Numerical results demonstrate the significant energy savings of the proposed scheme.