In recent years, there has been an increase in the development of medical robots to enhance interventional MRI-guided therapies and operations. Magnetic resonance imaging (MRI) surgical robots are particularly attractive due to their ability to provide excellent soft-tissue contrast during these procedures. This paper describes a novel design for a tendon-driven gripper that utilizes four shape memory alloy (SMA) spring actuators and variable stiffness joints controlled by SMA coils for use in MRI surgical robot applications. The contact force of the gripper link is determined by the mechanical properties of the SMA spring actuators (SSA) and the angle of each linkage, and the joint stiffness can be adjusted by varying the electrical current applied to the SMA coil. To enhance the efficiency of the SSAs, a new cooling system using water has been proposed and implemented. To validate the effectiveness of our proposed gripper, we conducted three types of experiments, namely, a single SSA experiment, a single SMA coil experiment, and a whole gripper experiment. The experimental results demonstrate that the proposed water-cooling system can effectively solve temperature issues of SMA, and the joint stiffness in the austenite state is higher than that in the martensite state. Moreover, our experiments show that the presented gripper is capable of grasping and holding objects of various shapes and weights.