There was another thread where the name got much discussion. I hesitate to guess what the etymology of each word is but growing up in the Scottish lowlands I can say that brae usually meant a road down a steep slope and not just the slope itself such that many roads were referred to names such as Bogs Brae or the Electric Brae. I'm not aware of there being any need for there to be water at the bottom of the brae as suggested by Wikipedia but then water generally gathers at the bottom of a slope.
Stoatin is probably more of a slang word as Brian suggests but not sure whether it is just confined to the Glasgow area or used further afield in Scotland. I suspect the latter. Either way, where I'm from it can mean great, as in very very good, but not great in a reference to scale ie. very very large. Stoatin can also mean hitting, or more commonly "stoat" or "stoated" for hit.
That said, what a bloody awful name for a course. I cringe whenever I tread the name and unreasonable as it sounds, I think I'd avoid playing the course just because of the name and irrespective of how good it is.
Niall