This paper proposes the integration of non-orthogonal multiple access (NOMA) and broadcasting with buffer-aided (BA) full-duplex (FD) relaying to improve the spectral efficiency of a single-source two-user network. Furthermore, opportunistic relay selection is employed, activating one or two FD relays to support the end-to-end communication. The network performance is improved through buffers, allowing to switch to half-duplex (HD) transmission/reception when FD operation is not feasible. A hybrid BA FD/HD NOMA relay selection algorithm is proposed, herein called BAHyNOMA, orchestrating the three aforementioned techniques, towards a simple yet efficient and robust relay selection. BAHyNOMA is evaluated in terms of outage probability, average sum-rate and average delay, and it is observed that the average sum-rate can be doubled, while maintaining low latency.