John,
The two that spring immediately to mind are The Honourable Company of Edinburgh Golfers and Notts GC, whose courses are known as Muirfield and Hollinwell. In both cases this is quite formal, and at Notts we even use Notts GC (Hollinwell) on the website, letterheads, documents etc. I believe in both cases this is because the club has changed course several times over their history so I imagine the use of a second name started to differentiate between the old and new course over which they play?
In Scotland there are many course that have several clubs attached which will mean two names associated with a course depending upon which club you are playing.
Courses with a formal name often also go by their local name ie the town or village nearest to the course, both locally but also to those who have an affection for the course:
Royal West Norfolk - Brancaster
Royal North Devon - Westward Ho!
Royal Cinque Ports - Deal
County Louth - Baltray
...to name just a few.
The previously mentioned Hoylake is the same but I'm not sure in all these case how official or formal the second name is? I suspect several will now use both names on websites for instance as some people may only be searching for the course under its informal name and they will need to utilise this to get more hits via search engines?
Cheers,
James