What would you call the small white cylinder in the bottom left hand corner of the picture? It may say "602" on it, and lights up purple with an annoying buzz/clicking sound in my particular tv.
But mostly what is it called so i can google-fu it?
That is a gas filed spark gap. It is designed to arc at a predetermined voltage to limit the voltage present to the rest of the circuit. The buzzing and purple light inside are from the internal arcing. if it is doing this all of the time, it could have become leaky and is firing at too low a voltage, or something else in the circuit is FUBAR, Possibly the florescent back light is bad. The whole ballast may just be shot, too.
ETA: After further contemplation, that may be a switching spark gap. If that is the case, it should fire to ignite the florescent tube. If it keeps firing, but the tube doesn't light, I would suspect the tube.