One of the settings for the backlight is to turn it OFF. So first, check the iPod's Settings screen to make sure it did not get set to be OFF, unintentionally. From the main iPod screen, select Settings. On the Settings screen, the location of the Backlight setting may depend on iPod model. On my iPod, it is under General.
If that's not it, be sure to try resetting the iPod, which is this procedure.
This basically "reboots" the iPod. The iPod is NOT erased or otherwise changed.
If the problem continues, try Reset Settings. This resets all settings on the iPod to their default setting. On the Settings screen, scroll to the bottom and select Reset Settings. The iPod is NOT erased. If the backlight is working again, you can set up your iPod as desired.
If that does not help either, you can try a Restore. This is done from the iPod's Summary tab in iTunes. This procedure WILL erase the iPod, reinstall the latest software, and set it to default ("factory") settings. If the backlight is working again, you can set up your iPod and sync music back to it, as desired.
If none that works, the most likely explanation is that the LED that provides the backlight has become faulty.