Optical quantum memories are important components in the long-distance quantum communication based on quantum repeater protocol. To outperform the direct transmission of light with quantum repeaters, it is crucial to develop quantum memories with high fidelity, high efficiency and long storage time. Recently, we demonstrate that it is feasible to achieve a high storage efficiency of 92% for electromagnetically-induced-transparency (EIT)-based memory with weak coherent signal pulses in cold atomic ensembles . To realize the highly-efficient memory with quantum light, we have built a bright and narrowband photon-pair source which can be locked to atomic transition based on the cavity-enhanced spontaneous parametric down conversion . Here, we present our results on the storage of single photons generated by such a source in EIT-based memories. A storage efficiency of 36% is obtained in initial runs. Future improvements toward a high efficiency are discussed. Such a development paves the way for the applications of photon-pair-based quantum repeater and multi-photon synchronization.