As many projects start, I found random parts that were laying around my room, and bought a couple additional things.
The parts are listed in the photo, but here are a few notes to help explain things:
I used an Arduino Duemilanove. You can use any Arduino as long as there are 7 digital (3 of them PWM), and 1 analog I/O ports.
The control knob I used fits the 1/4" shaft of the potentiometer. Make sure you buy a knob that has a set screw so that the baby can't easily pull it off the toy.
The resistors you need depend on what LEDs you use, what brightness you want, and desired buzzer loudness, etc. For the red 3mm LED (used for on/off status) I used a 390 ohm resistor. I used a 330 ohm resistor in the buzzer circuit so that the 'buzz' wouldn't be extremely loud. I used a 10k ohm resistor for the pushbutton switch and the rocker switch. For the yellow LED I used a 300 ohm resistor, and for the RGB LED I used 770 ohm resistors because the LED was super super bright.
I used 3 screws to attach the Arduino board to the case, 2 screws to attach the buzzer and 4 screws to attach the back plate. Screws work nicely because the threads are big and grab into the plastic case. Use whatever sizes fit your components.
Everything should be available for purchase at RadioShack, although they're also available at higher quality stores.
For tools you will need a soldering iron and 3D printer. If you don't have a 3D printer you can mount the switches into a box or send your file to a company that can print the case for you. You'll also need odds and ends like a screwdriver, wire cutters, solder...
Here are links to the parts I used: Buzzer Momentary pushbutton switch Rocker switch Control knob 3mm red LED Arduino Duemilanove
RGB LED Rotary potentiometer Yellow LED 5mm 9v battery connector Toggle switch Resistors Wire Header Heat shrink Replicator 2 3D printer Solder station