This weight, or mass, serves to dampen any tendency of the rudder to "flutter," or vibrate rapidly from the effects of aerodynamic forces on the control surface. According to Aerodynamics for Naval Aviators, a highly regarded text that looks at aerodynamics and aircraft engineering from a pilot's perspective, the distribution of mass and stiffness in a structure, such as a rudder, determines certain natural frequencies and modes of vibration. Aerodynamic forces can interact with the structure to excite or negatively damp the natural modes of the structure and allow flutter. Flutter is a bad thing - the high-frequency vibrations can rapidly lead to failure of the structure.
A simple way to change the natural frequencies and modes of vibration of a control surface by altering the distribution of mass is to place additional mass ahead of the hinge line. That's exactly what Piper did in the early 1970s with its single-engine Comanche models. In response to a perceived problem with vibration in the rudder, Piper designed a mass balance device that could be attached to the top of the rudder. Though inelegant in appearance, it's an elegantly simple engineering solution to a potentially serious problem.