Photonics offers a route to fast and distributed quantum computing in ambient conditions, provided that photon sources and logic gates can be operated deterministically. Quantum memories, capable of storing and re-emitting photons on demand, enable quasi-deterministic operations by synchronizing stochastic events. Interfaced source–memory systems are thus a key building block in photonics-based quantum information processors. We discuss the design of the single-photon source in this type of light–matter interface and present an experimental system based on a Raman-type quantum memory. In addition to the spectral purity of the produced heralded single photons, we find that their temporal distinguishability also becomes important due to the implicit temporal binning derived from photon storage in the memory. When aiming to operate the source–memory system at high repetition rates, a practical compromise between both of these requirements needs to be found. Our implemented photon source system demonstrates such a solution and enables passive stability, high brightness in a single-pass configuration, high purity as well as good mode matching to our Raman memory.