This page is for simple mods without buying backlight panels specifically designed for the gameboy. If you do need a professional mod, ebay or nonfinite should suffice.
Simple tutorial from Der-Warst. Extremely simple mod, however your mileage may vary. (Be careful with this one, if you use an LED that has different power requirements to the one that is already installed (read: Probably) then you may cause damage to your LED and gameboy.) 
Slightly harder backlight project by Bibin, using backlight from broken Nintendo DS. This mod should also work for the Gameboy DMG, however you will need to alter the resistor used, and connect it to where the battery indicating LCD is.