This problem is, as you suspected, almost certainly a problem with your monitor's backlight.
However, there are a number of possible causes for a backlight problem besides a bad fluorescent bulb. The bulb must receive a good supply of power in order to operate and that power is controlled via a number of circuits.
By far the most common cause of backlight failure is a problem on the POWER SUPPLY circuit board. Poor quality components, overheating, or just old age will lead to component failures on the power supply circiut board, usually starting with the electrolytic capacitors. Bad capacitors are sometimes obvious to spot in that they are often scorched, have bulging tops or are leaking dark gunk from the inside. However, also be aware that some failed components have no outward signs at all.
If you are handy with a screwdriver and have a basic knowledge of electrical safety precautions, then you can just disassemble the unit and replace the power supply board yourself with a new one from EBay. Here is a link to one that is available for your model:
Just stay away from ay that say AS IS in their description of condition, this is EBay-speak for broken. Make sure it says TESTED WORKING or something similar in the desription and you should be OK.
If you are really handy with a soldering iron, you can actually order just the capacitors and replace the components yourself. Here is a link to a good supplier of repair kits for that model of monitor:
The problem could also be with the INVERTER CIRCUIT but most often those types of failures will not be intermittent as you describe.