First guess: dying battery.
Have you got a battery charger of any kind? Try this:
1) Remove all connections to the battery. Don't need any possible electrical problems in boat wiring interfering with this test.
2) Pop the cell caps and check water level; fill each cell to about 1/8" to 1/4" below the notched fill tubes with distilled (and ONLY distilled) water. Replace caps. Careful, that's sulfuric acid in there! If the water level is way low, it may be that it's too late to save the battery (charging boils of water, it has to be replaced regularly).
3) Charge it overnight about 10 amps (or a full 24 hours at 2-5 amps, better), disconnect charger, wait a minute or two, get a voltage reading on battery with a meter. If that reading is below about 12.5-12.6v, the battery won't take a charge, it's dead.
4) Let the battery sit undisturbed, unconnected to anything, for another 24 hours. Get another voltage reading. If that reading is below about 12.4-12.5v, the battery is dying.
5) Even if the second reading is still OK, the battery may still be on its last leg. After the charge-and-sit routine, take it to an auto parts store and have them load-test it. That'll tell you if the battery has any capacity left. Load test might cost a few bux, but they'll probably credit that to the cost of a new battery.
In my experience, private sale boats are, coincidentally, sold at right about the same time they need new batteries