You didn't provide any really important details to help with the diagnosis. However, the first point is that if the starter motor will not turn the engine over the problem is not the main EFI relay. The main relay is completely unrelated to the starter motor system. In the absence of more details I suggest three easy places to start
1) The battery cable clamps on Honda products are very light weight and easily damaged. This results in poor connections at the battery. Go out to the front and see if you can twist the clamp on the battery post. If the clamp twists the clamps need to be fixed. Re termination with new clamps is best; but, battery post shims can provide a temporary fix.
2) How old is your battery? Modern maintenance free batteries can go from hero to zero in about 1 minute with little advance warning. Car runs fine, shut it off and come back 5 minutes later and it won't start (been there). Turn the headlights on. If they don't come on or die as soon as you try to engage the starter chances are that it is the battery.
3) Did the button that operates the clutch interlock crack and drop out (another chronic NSX problem) which blocks the starter motor form engaging? Little pieces of plastic in the driver's foot well that were not there before are a tell tale sign.
The diagnostic details that are important are what happens to the headlights while you attempt to start the car and what is going on with the dash lights. If you turn the headlights on and they stay bright when you attempt to start the engine then the battery and the battery clamps are probably OK. The more likely culprit is the ignition switch (providing the clutch interlock button hasn't dropped out). If you turn the ignition switch to the run position and the dash warning lights do not light up then chances are that it is the ignition switch; but, always go through the service manual test to check the switch to confirm that it is the cause of the problem.
If your car is a 1991 and you are still running on the original ignition switch just replace it even if it isn't the problem, because like the main EFI relay, it will eventually fail.
