Before investing in a esr meter, you'll want to investigate the CCFL tubes.
This behavior you see is often caused by a poor contact between the wire and a ccfl tube, or a ccfl tube that's too old and drawing too much current.
In both cases, the inverter tries to start the lamp pairs and it monitors the actual current consumption and whether the lamps actually run or not. If there's a poor contact at the end of a cclf tube, a pair of lamps won't start and then the inverter turns itself off.
If one of the lamps is going bad due to age or some manufacturing problem, the lamp will pull more current from the inverter which can damage the transformers and inverter circuit, so the inverter IC kills the backlight.
You'll have to open up the back panel and check the silicone ends of the ccfl tubes and the wires and if the cable is just loose, resolder it. If the ccfl tube is bad, you'll probably have to replace both tubes at same time (if not all four, if you don't want the other pair to go bad in a year or so) and they cost about 10$ each, plus shipping.
See this gallery for examples of desoldered/broken wires :
http://s807.photobucket.com/user/budm/media/Bad%20backlight%20lamp/DSC01525.jpg.html?sort=3&o=4And this gallery shows examples of bad ccfl tubes :
http://s807.photobucket.com/user/budm/media/Bad%20backlight%20lamp/DSCF1700.jpg.html?sort=3&o=22