Cool, sound like an interesting project. Before I start to suggest anything, first we need to know what is the type of facilities and tools available in your university. An advance CAM machine with higher DOF would help to fabricate advance and customized component, anything that can help you to achieve desired grip mechanism and size.
Then is the type of material you want to use. Assuming budget is not a problem, ideally would be an aluminum body, with high torque servos, and a mini replication of two robotic arm, comes with 5DOF/each.
But that would be too fancy for a Rubik solver, won't it
Rubik comes with 6 faces; to grip the cube at any given time, your grip need to be in contact of at least 2 faces, directly opposite each other. The cube has 6 rotating faces; of which 2 faces is in direct contract with the (first) grip. Understand this, and how you wish to rotate entirely depends on what type of resources you have to fabricate the arm.
Motor should be the last to consider. You can have servos, or more powerful stepper, or even DC motor; the later has less accuracy and control, hence your solver will have more rugged movement.