A foot-driven rehabilitation mechanism is suitable for home healthcare due to its advantages of simplicity, effectiveness, small size, and low price. However, most of the existing studies on lower limb rehabilitation movement only consider the trajectory of the ankle joint and ignore the influence of its posture angle, which makes it difficult to ensure the rotation requirements of the ankle joint and achieve a better rehabilitation effect. Aiming at the shortcomings of the current research, this paper proposes a new single degree-of-freedom (DOF) configuration that uses a noncircular gear train to constrain the three revolute joints (3R) open-chain linkage and expounds its dimensional synthesis method. Then, a parameter optimization model of the mechanism is established, and the genetic algorithm is used to optimize the mechanism parameters. According to the eight groups of key poses and position points of the ankle joint and the toe, the different configurations of the rehabilitation mechanism are synthesized and compared, and it is concluded that the newly proposed 3R open-chain noncircular gear-linkage mechanism exhibits better performance. Finally, combined with the requirements of rehabilitation training, a lower limb rehabilitation training device is designed based on this new configuration, and a prototype is developed and tested. The test results show that the device can meet the requirements of the key position points and posture angles of the ankle joint and the toe and verify the correctness of the proposed dimensional synthesis and optimization methods.