As an enabling technology for vehicle-to-vehicle (V2V) networks, multi-access edge computing (MEC) provides a feasible platform for sharing power and resources, and offloading some of the computation-intensive tasks between vehicles. This, however, is challenging with the unpredictable variations in road traffic conditions and vehicle mobility in MEC-enabled V2V networks. Consequently, such computation task offloading can be easily disrupted, which may require frequent switching of task offloading between vehicles and degrade the Quality of Service (QoS). In this paper, we focus on the computation offloading problem under unstable connections in MEC-enabled V2V networks. We first model this as a distributed online service optimization problem, which is proved to be NP-hard. In order to minimize the out-of-service time (i.e., the service mismatching, switching and compromise time), we propose a distributed Online Instability-aware Computation Offloading (OICO) heuristic algorithm to improve the service efficiency and quality. Specifically, in order to minimize the service mismatching rate, we design an efficient Service Path Matching (SPM) algorithm for matching pairs of customer vehicles (which require offload computing services) and server vehicles (which provide edge computing services) that share the longest matching path. We evaluate OICO through real-world traces, i.e., GAIA open dataset from DiDi. Extensive simulation results demonstrate that OICO can increase the service matching rate by 25% and reduce the power consumption by about 54% per customer vehicle compared with the existing schemes.