Mobile edge computing (MEC) technology has emerged, as processing a tremendous amount of data information has gradually become an urgent need in various industries. In the case of many land obstacles and inconvenient installation of MEC servers, a scenario where unmanned aerial vehicles (UAVs) relay assist users to offload tasks to the base station is proposed. Aiming at this scenario, an optimal task offloading method based on game theory is presented. In order to minimize system delay, the task-offloading ratio and task-offloading strategy are jointly optimized. Since these two variables are coupled with each other, the original optimization problem is transformed into two sub-problems for solving. First, the conditions for the existence of the minimum system latency are proved under the determined strategy. Simultaneously, the closed-form expression of the optimal offloading ratio is given. Then, the optimization problem is transformed into a task assignment problem and a game theory model is developed. By proving the existence of Nash Equilibrium (NE) in this model, the set of user task offloading policies based on the minimum delay is given after several iterations. The simulation demonstrates that the global computation time delay is effectively reduced by the above method, which is better than some other common offloading methods in terms of timeliness.