Engine quits running

I have a 2005 Mercury 40 HP bigfoot,4-stroke with EFI. I use non-ethanol gas in it. Usually runs great for trolling or at high speed BUT while on the water the other day it just quit while running about half throttle. I tried to restart it but didn't even get the "beep" when turning the key. Waited a few minutes then it started again and ran for about a minute then quit again. Seemed like it just lost power. I checked the oil, plug wires, fuel, temp, etc. Started again after a few minutes and it ran fine for about 15 minutes then started just quitting again. This happened 6 or 8 times this day. Ran it the next day at medium to full throttle for an hour with no problems. Does anyone have an idea about what could be happening here? I would hate to get stranded!!

Check connector where it plugs into ECU