If I ever became fantastically wealthy, with disposable income enough to build my own golf course, it would be The Martlet Club, an homage to the mascot for my school's sports teams. Faculty, students, alumni, and families of students and alumni would comprise the membership.
The logo: a Martlet with crossed clubs.
A few words on golf course names:
There seem to be four popular, general forms by which to name golf courses, and most interestingly, they seem to correspond (albeit very loosely) to different levels of quality. They are:
1. ________ ________ Golf Course--Almost always public, often municipal courses. Quality-wise, lowest of the four distinctions; quality ranges from bad (most dumpy courses) to good/very good (e.g. Wintonbury Hills Golf Course, Bethpage Black Golf Course)
2. _______ _______ Golf Club--Can be public (e.g. Whistling Straits Golf Club) or private (e.g. Shinnecock Hills Golf Club), and range in quality (Gillette Ridge Golf Club gets my vote for worst of the bunch), but overall a bit higher than _____ _____ Golf Course courses.
3. ______ ______ Country Club--More often Private than not; better than "Golf Club" courses on the average (Pine Valley Country Club), but a bad "Country Club" tends to be rarer than a bad "Golf Course" and "Golf Club"
4. (The) ______ ______ Club--Almost always private, and almost always of higher quality than the previous three (but again in a wide range, of course). Exclusivity is often higher as well (The Creek Club, Cypress Point Club).
Now of course, these four categories don't account for everything, andmay not quite always hold true, but I think it's worth keeping in mind in a general sense, at least. To demonstrate, You might think of one familiar example for each near your home, and rank the courses. More often than not, I'll bet the "Golf Course" will be last, and the "Club" may well be first.
Here's an example for Connecticut:
4. Wintonbury Hills Golf Course
3. Hartford Golf Club
2. Country Club of Fairfield
1. Stanwich Club
It's just an example (I'll bet that whether CCF is better than Stanwich or not is debatable, but you get the idea...), but I'd imagine that it can be said in a lot of places.