In this paper, we address the antenna selection (AS) problem in full-duplex (FD) cooperative non-orthogonal multiple access (NOMA) systems, where a multi-antenna FD relay bridges the connection between the multi-antenna base station and NOMA far user. Specifically, two AS schemes, namely max-$\SUu$ and max-$\SUuu$, are proposed to maximize the end-to-end signal-to-interference-plus-noise ratio at either or both near and far users, respectively. Moreover, a two-stage AS scheme, namely quality-of-service (QoS) provisioning scheme, is designed to realize a specific rate at the far user while improving the near user's rate. To enhance the performance of the QoS provisioning AS scheme, the idea of dynamic antenna clustering is applied at the relay to adaptively partition the relay's antennas into transmit and receive subsets. The proposed AS schemes' exact outage probability and achievable rate expressions are derived. To provide more insight, closed-form asymptotic outage probability expressions for the max-$\SUu$ and max-$\SUuu$ AS schemes are obtained. Our results show that while the QoS provisioning AS scheme can deliver a near-optimal performance for static antenna setup at the relay, it provides up to $12\%$ average sum rate gain over the optimum AS selection with fixed antenna setup.