In many remote areas lacking ground communication infrastructure support, such as wilderness, desert, ocean, etc., an integrated edge computing network in the air with edge computing nodes is an effective solution. It can provide over-the-air computing services for ground devices (GDs) with limited computing resources and battery life. In this paper, we study task offloading and resource allocation in the aerial-based mobile edge computing (MEC) system supported by a high altitude platform (HAP) and unmanned aerial vehicles (UAVs), with the goal of minimizing the GD's energy consumption. Considering that the task arrival of GDs and wireless communication quality are both stochastic and dynamic, we apply stochastic optimization techniques to transform this task offloading and resource allocation problem into two subproblems, i.e., 1) a subproblem for local computation resource allocation and 2) a subproblem for offloading resource allocation. For the first subproblem, we use convex optimization methods to address it. For the second subproblem, we use game theory to formulate the competition of offloading resources among GDs and propose the Distributed Game-theoretical Multi-server Selection (DGMS) algorithm and the Transmission Power Allocation (TPA) algorithm. Finally, we propose a Distributed Online Task Offloading and Resource Allocation (DOTORA) algorithm and give the theoretical performance analysis of the algorithm. We perform extensive experiments, including the comparison experiments with the UAV-Only and HAP-Only framework, and the comparison experiments with other algorithms under our HAP-UAV framework. The experimental results validate our proposed framework and the DOTORA algorithm.