Is it possible to fit a blue LED under the bridge of my RG1570... ? If yes, then from where will the LED take power?
My Guitar Rig:

> '09 Ibanez RG1570 Mirage Blue w/ DiMarzio John Petrucci Set
> ESP LTD V-50 Black
> Line 6 POD X3 Live
> Dunlop DB-01 Crybaby From Hell
you could have the battery with all of the pots and switches.

run the wires through the hole that the bridge ground goes through
You'll need to get a battery because there's no way you can get power another way cleanly. You can get watch batteries which won't take much room. You might be able to slip a rechargeable aaa in the spring cavity.

Undo the backplate and see. Led's aren't massive and there should be room. Don't forget a resistor (and a switch if you're fancy). Maybe even set it to autolight when a jack is plugged in.

Now if you were really, really fancy, you could wire up a small circuit and give it a slow on/off glow, but you'd need to plan it carefully.
There's a good chance that what I've written above is useless and if you take any of the advice it's your own fault.