As a privacy-preserving solution, federated learning (FL) demonstrates great potential in distributed model training, but limited bandwidth, particularly in near-field communication (NFC)-based systems, emerges as a key bottleneck by restricting the number of participating clients. To address this challenge, over-the-air FL leverages the superposition property of wireless multiple-access channels, enabling faster model training and accommodating more clients, even in bandwidth-constrained scenarios like NFC. However, due to its analog-integrated nature, the FL performance is also affected by other factors, such as channel noise. These motivate us to consider how the selected client set and channel noise affect FL performance. To explore this concern, in this article, we consider an over-the-air FL system with analog gradient aggregation and analyze the impact of the selected client set and channel noise on FL training performance. The theoretical analysis effectively shows the importance of the clients' number and the power scaling factor to the FL training performance. Based on the theoretical analysis, we transform the global optimization problem into the client selection problem and propose a dynamic client selection scheme to optimize the training performance under the aggregation error constraint. Experimental results demonstrate that our proposed scheme can boost FL by speeding up the convergence of the global model (at least 35%) and saving energy consumption.