The backlight for the LCD screen in most laptops is a cold-cathode fluorescent lamp, which is basically just a small fluorescent tube. Just like fluorescent room lighting, they eventually burn out. Unlike room lighting, they are not made to be replaceable.
Here's my quick and dirty (and cheap) project to replace the lamp on my old laptop, who's backlight burnt out recently. I expect you could make this look a lot better if you were willing to put some more money into sourcing parts.
Materials: Screwdrivers, Dremel, electrical tape, wire crimping tubes.
The specifics of disassembling any particular laptop are going to vary widely, so I'm going to skip that step for simplicity. If you're not confident just diving in to your laptop, you'll frequently be able to find illustrated parts breakdows (IPBs) at the manufacturer's site for some guidance.
NOTE: High resolution photos available on Flickr: http://flickr.com/photos/killerrobotclan/sets/72157605872353185/detail/
Step 1: Getting at the Lamp
Though the specifics of how to do this will vary widely, you'll need to entirely remove the LCD assembly from the lid of your laptop, so that you can get at the bottom edge which houses the lamp. It should be the only part of the screen which has heavy gauge high-voltage wires going to it. It will be connected to a shielded inverter circuit, which is often inside the laptop itself, but sometimes behind the screen, as here.
I bought a generic white CCFL tube at a locally owned hobby shop; these are also available online. The easiest way to test that the laptop's inverter would light it correctly was to hook it up. I used simple wire crimps to splice the wiring tail from the old lamp onto the new one, and plugged everything in. Looks good so far.
Step 2: Electrical Tape, Part 1
Since the backlight wires carry high voltage, we want to make sure they're taped up anywhere they might be exposed.
Sine the diameter of my replacement lamp is too big, it won't fit into the original mounting assembly, so I'm securing it with electrical tape.
There's a large flat diffusion layer behind the LCD that diffuses the light from the lamp evenly across the screen. We want the lamp to be directly illuminating this diffusor, rather than the LCD itself.
Step 3: Testing
Lets flip the screen over and start up the computer to see how we're doing. We immediately see the consequences of having the wrong diameter CCFL tube; light leaks past the diffuser and directly into the LCD near the bottom, causing the oversatuation at the bottom edge. Still, it's working.
Step 4: Reassembly, Part 1
The cable exits on the new lamp are too wide, so we have to Dremel small sections out of the frame and chassis to accommodate.
Step 5: It Works!
It's quite messy, but it was cheap and saved a laptop that was about to be thrown away, so I'll still call it a success.