First thing, would be helpful if you gave us the last four-three numbers of your Serial number, to pinpoint what's your machines.
Thus said, the average causes of a dim or flashing backlight are, in order of probability:
1. An aging inverter
2. A worn out inverter cable, damaged and/or frayed inside or with loosened connectors
3. An aging CCFL, the "neon-like" lamp lodged inside the LCD panel
4. (Very, vary rare) The capacitors on the Logic Board responsible for sending power to the Inverter through the Inverter Cable losing power.
You're sure your logic board is in perfect working order, and usually it is, so put aside that bad thought.
The inverter you have is fresh, as fresh is the cable.
First, check out if the cable fits snugly, for example by moving up and down the monitor, with slow movements and steady hand.
If you're sure the cable is not only working, but with no loose connection, and the inverter is fresh, the only other possible culprit is the CCFL lamp.
However, the CCFL lamp is a thin and long neon-like lamp lodged in the LCD panel, wrapped by an aluminium foil that acts as a reflector, spreading the light evenly on the whole monitor.
A new CCFL lamp costs roughly $10-15, a fresh LCD panel ranges from $120 to $200 new, the refurb half-priced.
However, changing only the lamp involves a whole lot of work, that mostly can be summarized in peeling the aluminium foil, avoiding to crimple or tear it to preserve the even reflection of the light, desolder a couple of soldering points, and resolder the new lamp, and then replacing the foil and hoping to have not messed somewhere in between.
Thus, many individuals just get a new panel, refurb or new, and swap it in.
So, my advice is: recheck throughly the inverter cable and the connection between inverter and monitor.
If you're sure it's all fitting good, you've narrowed the issue to the CCFL lamp inside the LCD.