1300AE
According to my '06 service manual the neutral indicator light draws it's power from the windshield drive unit, which gets it's power from the signaling system fuse. It goes through the light emitting diode (LED) in the instrument cluster, then travels to both the neutral switch (which I believes lives in the engine), and the engine control unit (ECU). The fact that you are getting light rules out the fuse and wires leading up to and including the LED.
This leaves the wires leading away from the LED. This is the path back to battery negative terminal. The neutral switch is a mechanical contact which should be either on or off. Since it lives within the engine (actuated by the shift drum) I would believe it to be working fine. I don't expect it to be RPM sensitive. This points back to the ECU, which keeps track of, among other things, engine RPM. This would be the engine speed sensitive item. Hopefully it is something as silly as a fault in the grounding buss - i.e. the grounding spiders.
Have your dealer investigate the usual suspects, the S4 spider nest living under the fuel tank left and forward of the engine; and the S6 spider nest which lives near the left turn signal in front of the glove box. The easy way to get to that is to remove the black inner cowling panel next to the left horn and radiator. Any sign of overheating in the white plastic connector or black cap would be the smoking gun. If there were a high resistance at these spiders the amperage would find other routes to get back to battery negative terminal. This may be causing the ECU to behave in this manner.
If this is the smoking gun that is causing the problems, have your dealer * fix it first. Then I would suggest installing either my Grounding Harness which replaces all 6 spiders with a secondary path (fat 10 gage copper wire) directly back to battery negative terminal, or my good friend Art Cooper's (roadrunner) harnesses which Tee in to the high amperage devices pigtails (headlight and radiator fan) and divert much of the current to an engine cover screw - which eventually makes its way to the battery negative terminal. Both systems will do the job of keeping the yamaha grounding buss from being overwhelmed in it's old age.
It's no guarantee that this is the problem, however, it can't hurt by providing a good ground on these "
electrically engineered challenged" 2nd generations FJRs.
Hope this helps.
Brodie
* By the way, I have several hundred of the male connectors with the pins. I am thinking about making some spider nest repair kits for the mechanically inclined.