I am far from a monitor expert, but it sounds to me like the CCFL tube or power inverter is going out. My understanding is this is pretty much like a ballast on a fluorescent light. It pumps out a high jolt of power to light up the ccfl tube to light it, Older tubes don't turn on like they should right away, and the inverter no longer attempts to jolt it after a few seconds.
I have fixed a monitor with this issue, and it pretty much involved decasing the monitor, sliding in a new after market ccfl tubes on the top and bottom of the lcd panel, and wiring it to the inverter if the inverter is still good. Not too hard, but if the inverter is going out, you can always just externally wire up a second inverter to your monitor. I just used a inverter that plugs into a psu and never even bothered wiring it directly into the monitor, just had another wire going to the computer.
Also, don't worry if the tube size is an inch or so shorter than your old one. They charge a lot more for monitor precise sizes, and a shorter tube lights up just fine.
Oh, and I learned the hard way that ccfl tubes are very fragile. Don't bend. And don't touch the end of the tube either. Youch!