According to the Internets the best way to tell if it's the inverter board or the CCFL that's failed is to swap one out for a known good replacement - usually the inverter board as that's easy to get at. This does assume you have a spare inverter of the right sort to hand... which as it happens, I do have as I've got the previous hardware for Aether, a ThinkPad R50e I had at university. The R50e and R50p are essentially the same hardware, just with different options fitted. The inverter boards aren't the same FRU number but both have the same connectors, and the board for the R50e even has the WiFi and Bluetooth LEDs. So, wrestle with the R50e to get the bezel off that (quite a bit easier, now that I know where the clips and glue are), swap the inverters, and power Aether back up.
And of course it didn't work.
Because being able to fix a dead LCD by transplanting an easy-to-access inverter board would be far too easy a fix. This means the CCFL tube is on its way out, and getting at that requires dismantling the LCD panel itself.
I think at this point I've got very few options for mending Aether. I could swap the hard disk back into the R50e chassis as that's still functioning, but I'd lose the onboard WiFi (which I only upgraded last week!) and would have to go back down to a 1024x768 screen. Additionally the screen in the R50e, while having a working backlight, has a column of blue pixels on the right-hand side stuck half-on. That's likely a failed connection to the panel itself and so falls squarely into the "not fixable" category.
Alternatively, since the panels are physically compatible I could swap the two LCD panels over. This would demote the R50e down to being purely a parts donor, but would get the R50p fully functional. However that still has the same screen-related downsides of the lower resolution and stuck pixels.
Or there's a third option, which is to replace the CCFL tube. It's both rather risky - you have to extract a long, thin, and above all fragile glass tube from the bowels of the LCD display without breaking it or getting dust inside the screen - and also requires a replacement tube of the right sort. The tubes themselves appear to be unobtainium these days as everyone's moved on to widescreen panels and the only eBay sellers with correct tubes are based in the US. However as it happens I do have a potential donor laptop in the form of the R50e. The LCD panels in both laptops are physically the same size and have the same wiring at least for the backlight - whether or not the tubes themselves are the same or close enough is unknown, but I reckon there's a good chance. This is easily the highest-risk option though as there's the very real possibility of breaking both LCD panels.
There's also the fourth option of a replacement LCD panel but that's more money than what I should spend on a 10-year-old laptop...
Hmm... well, the panel in the R50p is definitely unusable as is, so I don't really have anything to lose by opening up that one and seeing how practical the tube replacement is. The panel in the R50e, on the other hand, still works aside from the stuck pixel column, so I would be risking that... but to be honest, one of my primary reasons for keeping Aether going this long is the 1600x1200 screen in it. You just can't get laptop screens that large anymore - they top out at 1920x1080 and at that point you don't have a laptop, you've got a desktop with a built-in UPS. You can, on the other hand, easily get laptops with better displays than 1024x768.
I think it's worth a try...