There are really two possibilities that cause the backlight to die on the ipod gen 3..
1... Lcd went bad. Just like on a laptop, the backlight is part of the lcd assembly. easy to replace IF this is the culprit..
2... Motherboard went bad. The IC that is the driver chip on the motherboard for the backlight went bad. This, for most people is unrepairable. It is that small, and easy to screw up,.
The best way is to try a new lcd in the ipod.. Takes about 20 mins of work to get down to the lcd and removing it and trying a new one. There is an excellent tutorial on this board that will guide you if you wish, or take it to someone that knows how to do this. If it is the lcd, this is one of the simplest repairs on the ipod 3.. enjoy