Due to the limitations of computing resources and battery capacity, the computation tasks of ground devices can be offloaded to edge servers for processing. Moreover, with the development of the low earth orbit (LEO) satellite technology, LEO satellite-terrestrial edge computing can realize a global coverage network to provide seamless computing services beyond the regional restrictions compared to the conventional terrestrial edge computing networks. In this paper, we study the computation offloading problem in the LEO satellite-terrestrial edge computing systems. Ground devices can offload their computation tasks to terrestrial base stations (BSs) or LEO satellites deployed on edge servers for remote processing. We formulate the computation offloading problem to minimize the cost of devices while satisfying resource and LEO satellite communication time constraints. Since each ground device competes for transmission and computing resources to reduce its own offloading cost, we reformulate this problem as the LEO satellite-terrestrial computation offloading game (LSTCO-Game). It is derived that there is an upper bound on transmission interference and computing resource competition among devices. Then, we theoretically prove that at least one Nash equilibrium (NE) offloading strategy exists in the LSTCO-Game. We propose the game-theoretical distributed computation offloading (GDCO) algorithm to find the NE offloading strategy. Next, we analyze the cost obtained by GDCO's NE offloading strategy in the worst case. Experiments are conducted by comparing the proposed GDCO algorithm with other computation offloading methods. The results show that the GDCO algorithm can effectively reduce the offloading cost.