<p>Quantum key distribution is the art of sharing secret keys between two distant remote parties. However, owing to the imperfections in the practical devices, the eavesdropper Eve can apply some attacking strategies to affect the randomness of the basis selection in quantum key distribution systems. In this work, we prove the lower bound of the secret key rate for quantum key distribution systems from the perspective of basis selection hidden variables. Then, we analyze the lower bound of the secret key rate with the weak-randomness model. The growth of weak-randomness parameter will reduce the maximal tolerable error rate and transmission distance of quantum key distribution systems. To improve the performance of the practical quantum key distribution systems with weak-randomness basis selection, the advantage distillation method has been applied. We believe our work provides a tool for evaluating and enhancing the security of quantum key distribution systems with weak-randomness basis selection.</p>