Whatever operates the backlight is dying, what to buy to fix?

You talk about fidgeting with the backlight controls or you restart your system. When you adjust it does the display brightness icon show up with the setting low or where you set it last. If it wasn't I suspect your CMOS battery is dying. Here's the IFIXIT guide to replace it CMOS Battery Guide. One way to tell is if the date & time setting is wrong when you don't have your system connected to the internet (by default your system will correct its self from the internet time service). If you haven't replaced your battery since getting your system it's likely time to do it any ways.

OK, lets say the CMOS battery is not the issue then your CFL's or the inverter and cable will need to be checked out. Most of the time it's the inverter and the cable that goes through the hinge area that need replacing. Here's the IFIXIT guide on replacing the Inverter Inverter Guide and here's the guide on the Inverter cable Inverter Cable Guide.

If the CFL's are going, you may want to think about replacing the complete lid assembly as replacing the CFL bulbs is not that easy a job. You'll need to take the entire display assembly apart to get to the bulbs (which are fragile) and soldering the wires is a bit tricky.

Of course your skill set and how much effort each of these tasks take will need to be weighed against getting a new system.