You're looking at 3 different parts to this: the remote control, the physical arm, and the microcontroller that interfaces the two together.
As far as it being under 100 bucks, it all depends on what you already have. If you don't have access to the basic tools then forget about it. If you only need the servos and controller, you can absolutely do it.
Anyways, the remote control can be store bought or homemade, it doesn't matter as long as it works. As far as the arm is concerned, you're going to want three servos, one for each joint. Since all you want to do is wave around a light then strength shouldn't be too much of an issue. Everybody likes AVR for microcontrollers* so that should be easy to find tutorials on.
The big thing to remember is that you want to keep weight to a minimum (normally you also would want to maximize strength but a light on the end isn't very heavy)