This paper proposes a novel Hermite neural network-based second-order sliding-mode (HNN-SOSM) control strategy for the synchronous reluctance motor (SynRM) drive system. The proposed HNN-SOSM control strategy is a nonlinear vector control strategy consisting of the speed control loop and the current control loop. The speed control loop adopts a composite speed controller, which is composed of three components: 1) a standard super-twisting algorithm-based SOSM (STA-SOSM) controller for achieving the rotor angular speed tracking control, 2) a HNN-based disturbance estimator (HNN-DE) for compensating the lumped disturbance, which is composed of external disturbances and parametric uncertainties, and 3) an error compensator for compensating the approximation error of the HNN-DE. The learning laws for the HNN-DE and the error compensator are derived by the Lyapunov synthesis approach. In the current control loop, considering the magnetic saturation effect, two composite current controllers, each of which comprises two standard STA-SOSM controllers, are designed to make direct and quadrature axes stator current components in the rotor reference frame track their references, respectively. Comparative hardware-in-the-loop (HIL) tests between the proposed HNN-SOSM control strategy and the conventional STA-SOSM control strategy for the SynRM drive system are performed. The results of the HIL tests validate the feasibility and the superiority of the proposed HNN-SOSM control strategy.