Well, the culprits of a failing backlight are, in order of probability:
1. A failing Inverter Cable (a surefire to test it is moving back and forth the display in the hinges: if sometimes the backlight gets on, the Inverter Cable was pinched in the hinges and cut, otherwise, it may still be completely worn)
2. An aging Inverter
3. An aging CCFL (but in this case you also see your whites becoming pinkish and every other sign of a failing neon)
4. (but really rare) A problem in the capacitors giving power to the inverter.
So, yes, the best way to attempt a fix is replacing the Inverter + Inverter Cable. Is the most inexpensive fix, and solves the issue in the 90% of the cases.
Only if the issue still lingers, and only in that case, you may have to replace the LCD panel or the CCFL lamp. But keep in mind CCFL lamps are made to last, so... I believe the Inverter is busted.
Mine died recently too, after six years of honored service. Got it fixed in a jiffy. Using the iFixit guides, is fairly easy. Timeconsuming, but not impossible.