I agree with the two previous posts, the problem is likely the 12V supply to engage the solenoid. A wiring diagram for my boat shows small wire and eight connections between the battery and the solenoid. There are two end connections, two at ignition key, two at the start button, and two at an inline fuse. Any one of those being slightly loose or corroded along with too small wiring will cause your symptoms.
I solved this a different way. I mounted a $35. starter relay on the bulkhead next to the starter. This relay uses much less current. Have had no start problems since.
WM Link for "Solenoid-12V
Before the relay fix I had replace wiring, the key switch, the pushbutton, and had the starter and solenoid rebuilt. All to no avail!