In flying Internet of Things (Flying IoT) scenario, the relay technology can effectively solve the problem of timely communication with distant unmanned aerial vehicles (UAVs). It is practical to design a distributed approach to let UAVs self-organize their relay selection strategy while considering the impact of social networks of objects. In this letter, we first propose a social-aware relay selection scheme using matching theory, which takes both physical condition and inter-device social tie into account. Besides, we study how to find the unilaterally Pareto-optimal stable matching through inconspicuous manipulating the preference list. Simulation results demonstrate that the proposed scheme can strike a better trade-off among throughput, reliability, and computational complexity performance than other benchmarks.