Aside from a burned out bulb, one of the more common causes of a backlighting bulb not illuminating is that when a bulb is replaced the little copper contact "flap" gets folded under so it doesn't make good contact with the little black bulb holder socket:
Folded under - wrong:
Not folded under - correct:
A much less common issue is that there's a break in the circuitry somewhere so that not all of the backlighting bulbs are getting power/ground. The geeky way to fix that would be to chase the wiring in the blue tape in an attempt to find the break. A much easier way is to jump a wire from a working backlighting bulb to the non-working bulb. All you need to do is strip both ends of a piece of wire and run it from the working bulb to the non-working bulb. Take each stripped end and put it on the contact when you insert the bulb. In the example below assume that the upper bulb is non-working but works if it gets jumped power from the lower right bulb.
As with most DC circuits, you need to pay attention to polarity (+/-) when doing this. The diagram below shows the polarity of the backlighting bulbs.
