Could be the cdi box, rectifier, stator, coils, or just needs the coil wires replaced or trimmed a bit (yes it's possible, and needs solid core wire). It could also be a corroded connection in the electrical box. Has it ever been submerged ?
First thing is get a good agm battery for it, rebuild the carbs (might even look at getting a set of cdkII carbs instead of using the crap CV carbs), replace the oil injection lines and fuel lines.
Find a pdf of the service manual. It will have all the specs to check all the electrical components.
I would also get a GOOD compression gauge and do a compression check, but first put a tablespoon of 2 stroke oil down each plug hole and turn the engine by hand to coat the walls, rings, and pistons.
I have a ZXI manual and will try and post up some pages.