Recently, adopting UAVs equipped with the edge computing platform to provide computing service has been considered as a promising approach for resource-limited devices in mobile edge computing (MEC). Unfortunately, the limited resources (e.g., energy, computing and communication) of the UAV may significantly restrict its service capability, which means it has to selectively provide task offloading service to achieve the maximal benefit. In this article, aiming at optimizing the overall benefit of the UAV in a single dispatch, we propose an approximate Benefit Maximizing Task Offloading (BMTO) algorithm, which jointly considers the trajectory scheduling of the UAV and the offloading strategy of tasks. Specially, the flight path of the UAV is decomposed into several hover sites, which are selected by a benefit-cost approach. And the offloading sequence of tasks is arranged to maximize the benefit of the UAV through a surrogate function, which is proved to be a nonnegative monotone submodular function. Thus we transform the original problem into a submodular maximization problem and theoretically prove that BMTO owns an approximation ratio of 1/2(1-1/e). Simulation results show that our proposed algorithm outperforms the benchmark algorithms in terms of total benefit as well as energy efficiency ratio.