I specifically buy 3gs iphones with DIM screens (no backlite) to repair.
In my opinion, here are the most common causes:
1. Corrosion at #1 LCD connector on board and/or LCD
2. Bad LCD
3. 6R8 coil & chip/curcuit problem
The quickest test of backlite curcuit is:
Power on phone, remove screen, and do lite test at pin 3 of #1 LCD connector on board. If there is power...backlite curcuit is probably not broken. If not, troubleshoot backlite curcuit.