Is it powered off of your cranking batt (voltage draw causing it to cut out)?
Or is it powered off of your trolling motor batteries (loose wire causing a break in the circuit from the vibration)?
I'm betting it's the former. I'd look at a weak batt first. If not that, i'd check out the alternator with an electrical tester.
With all of the electronics on todays outboards, just about all of the alternators output at idle is consumed to by the outboard itself. There isn't much output above that to charge the cranking batt until the RPMs come up. Factor in the voltage draw from hydraulics (trim and jackplate) and you can sap the charge out of a batt pretty quick even with the engine idling. That's why most of the new motors require 1000 MCA minimum.