In the Industrial Internet, computing- and power-limited mobile devices (MDs) in the production process can hardly support the computation-intensive or time-sensitive applications. As a new computing paradigm, mobile edge computing (MEC) can almost meet the requirements of latency and calculation by handling tasks approximately close to MDs. However, the limited battery capacity of MDs causes unreliable task offloading in MEC, which will increase the system overhead and reduce the economic efficiency of manufacturing in actual production. To make the offloading scheme adaptive to that uncertain mobile environment, this paper considers the reliability of MDs, which is defined as residual energy after completing a computation task. In more detail, we first investigate the task offloading in MEC and also consider reliability as an important criterion. To optimize the system overhead caused by task offloading, we then construct the mathematical models for two different computing modes, namely, local computing and remote computing, and formulate task offloading as a mixed integer non-linear programming (MINLP) problem. To effectively solve the optimization problem, we further propose a heuristic algorithm based on greedy policy (HAGP). The algorithm achieves the optimal CPU cycle frequency for local computing and the optimal transmission power for remote computing by alternating optimization (AP) methods. It then makes the optimal offloading decision for each MD with a minimal system overhead in both of these two modes by the greedy policy under the limited wireless channels constraint. Finally, multiple experiments are simulated to verify the advantages of HAGP, and the results strongly confirm that the considered task offloading reliability of MDs can reduce the system overhead and further save energy consumption to prolong the life of the battery and support more computation tasks.