Federated Learning (FL) is a decentralized machine learning framework that supports collaborative model training with data privacy. Nonetheless, achieving fundamental stability—stable and predictable global model convergence across communication rounds—is still a great challenge in the presence of data heterogeneity, non-uniform client participation, and communication limitation. Current client selection methods, such as random and importance-based ones, tend to cause unstable model performance, particularly in non-IID data scenarios. This work proposes an optimization-driven client selection framework aimed at improving core stability in FL. Our method chooses clients by a quantified stability score, considering gradient divergence and update consistency to select participants most contributing to convergence. With the aid of this stability-aware selection scheme, our approach reduces performance instability, speeds up convergence, and improves robustness of the model. Comprehensive tests on benchmark datasets, such as CIFAR-10 and MNIST, illustrate that our method outperforms the state-of-the-art client selection methods. Our method produces more accurate results, shorter convergence time, and steadier model performance over training iterations. These observations highlight the imperative of stability-centered client selection in FL and lay a path to more robust and efficient federated learning models.