RGB LED strips - $16 x 4 = $64
Each strip is 1 m long. Most cars are less than 2 meters from the back of the front tire to the front of the rear tire. These strips can be cut to size, and wired together (end to end). They are coated in rubber, and so they are quite waterproof and shatterproof. They also come with a strong adhesive backing, that holds surprisingly well over time. At 12V, the LED's are bright enough for the glow to be seen well at night, and even a little bit during the day. Since these strips are RGB, they contain sets of red, green, and blue LED's - by mixing these three colors at the right ratios, you can create any color of glow (more on that later).
Arduino - $30
This popular, flexible microcontroller will be used to interpret user inputs, and control the LED strips accordingly. There are a ton of different variations
on the Arduino platform. The one I use is a bit of a home-brew solution - I actually re-purposed an old Sentry Controller
for the task.
MOSFETs - $1 x 3 = $3
MOSFETs are transistors that can switch a high-current load (the LED strips) using a 5V signal (from the Arduino). If you want to know how they work, read up on the wikipedia article
Voltage Regulator - $1
This will be used to step down the 12V from your car's electrical system to the 5V required by the Arduino.
Perfboard - $3
You can get this stuff at most hobbyist supply stores or online. We will use it to wire up the MOSFET part of the circuit.
Wire - approx 15 ft = $5
12 gauge should be fine - these LED strips can only take up to about 600 mA each. Get something waterproof and reliable, because it's likely going to be running on the bottom of your car - not a happy place for electrons.
Touch screen (Optional) - $10
I will show how to use a resistive touch screen (glass panel only, no display) to make a custom color selector. You may want to buy a breakout board
for the connector - otherwise it's going to be a pain to solder (I found out the hard way).
Single RGB LED - $2
This will go in the control panel. It's useful when adjusting the color from inside the car, so you know what color the glow is. Because it's not very suave to open your car door and look down to check.
Button - $0.50
This will come into play later, to ease the process of adjusting the color with the touchscreen.