Strange problem. First off the solenoid has to energize before any power is applied to the starter motor. On solenoids ready for replacement, due to high usage, the contacts pit and the surface area is reduced so when the starting current runs through there they actually weld together and releasing the key and de-energizing the coil that activates the solenoid, the current remains because the contacts can't disengage. So you turn the key to start, the solenoid closes, the starter spins and starts the engine....you release the key and it springs back to ON, not START, but the starter continues to spin while the engine is running. Read through that and that isn't your problem because you did not energize the starter to start the current that would have caused the welded contacts to weld in the first place.
So we go after how could the solenoid close if you didn't turn the key to START? First suspect is debris (from usage) inside the ignition key switch either floating around or making a conducting path (from usage) between 12v and the START wire terminal. Or the spring return inside the switch has failed and vibration is causing it to intermittently connect the START terminal to 12v. I'd suspect that first.....a bad switch.
Second would be a chafing in the wiring harness that allowed the above mentioned function to occur somewhere in the wiring harness from the control box to the solenoid.
Third would be a failure of the spring inside the solenoid that keeps the high current plunger from closing and supplying current to the starter.
At 7 years of age I would expect the key to be the problem especially if the boat was highly used or sat out in the environment unprotected. Wiring harness chafing could happen but for that exact combination of power getting to the start wire is highly remote. Having the spring go bad inside the solenoid is a never happen event if you ask me.
Thats the best I can do from here.