This article proposes a neural terminal sliding mode controller (TSMC) with finite-time (FT) convergence for the uncertain MEMS gyroscope dynamics. To address the uncertainty, considering the periodic tracking property of MEMS gyroscopes, a composite learning mechanism driven by the learning performance evaluation signal is applied to learn the system dynamics. By selecting the terminal sliding mode surface, the TSMC is constructed with error feedback and feedforward compensation. Under the TSMC with the composite learning, the system tracking can be guaranteed to be with FT convergence, while the weights of the learning system will converge in FT. The stability of the closed-loop system is analyzed by the Lyapunov approach. The highlight is that the design can obtain the learning knowledge while the information can be directly reused in repeated tasks with no need of online update. The effectiveness of the proposed method is verified via reference signal tracking of MEMS gyroscopes, while the controller using the stored knowledge achieves better tracking performance of faster convergence and higher tracking accuracy with no need of weight update.