2000 TRX300 starting issues

I am helping a relative with this atv. It will kick start and run good. You can use a screw driver to jump across the soleniod and it will start. I have noticed that the neutral light will only illuminate sometimes. I have not figured out why or if that has anything to do with it. Also sometimes the starter wil not stop. It just contiues to spin after the engine is running. I read the battery cable routing info posted by Shade tree and the cables are correct. The solenoid was replaced with a cheap one from ebay. We tried 2 of these ebay solenoids thinking one of them was faulty but the problem persists. Any thoughts? I was thinking I would put new brushes in the starter, This atv is in really good condition so it is worth the effort to repair.