There appear to be a lot of them. Is there something special a course needs to do to acquire this designation? Or, is it just an arbitrary way to name a course?
Note; Google shows there’s a National Golf Course Owners Association, but it does not appear to be an association of owners of “National Golf Courses”.
It’s just a name.
My impression is that some of these “national” courses, like Augusta National, are private courses which reserve a significant percentage of their membership slots for people who do not live near the course, but who visit occasionally. This brings in revenue without adding much to daily traffic, ensuring a very private golfing experience for all involved.
MY WAG would be that a “national” golf course fulfils certain conditions without which the course cannot serve as the venue for certain tournaments on a “national” level.
Nope. There’s no naming authority for this, and the PGA TOUR and USGA qualify courses to host their tournaments as necessary but these courses don’t add anything to their names once that’s been done. (At least, not in the US - golf is run differently in Europe and what you describe certain could happen if that’s they way it’s set up)
Crotalus does have a point that there’s probably some expectation that “National” golf clubs will use the entire country as its member pool, as opposed to selling full memberships locally and allowing those from outside the region to obtain cheaper, lesser memberships, but it’s certainly not binding.
Just curious – how many golf courses in the U.S. have nation-wide member pools and serve as “destination” courses? I can name only two for sure without looking it up – Augusta National and Pebble Beach. Maybe Sawgrass? But there’s got to be a bunch more.
Looking thru a couple of course guides for the Chicago area, I spotted:
Butler National (private - previous host of the Western)
Geneva National (semi-private)
George Dunne National (public- Cook County forest preserve - previously Forest Preserve National)
Plum Tree National (public)
So I think it is pretty clear that it is just a name.
Going one better, we’ve got Harborside International - a public course undoubtedly targetting the jetset! 
In my reading, I’ve been somewhat surprised at the number of folk who belong to multiple private clubs. Would mainly have to do with vacation homes and places of business, tho some is also status. There are plenty of status clubs out there. Cypress Point, Riviera, Oakmont, Pinehurst, Burning Bush (DC) …
In Chicago, I’d assume Butler, Medinah, and Olympia Fields have pretty large and national memberships.
How about Pinehurst? I’d call it a pretty solid “destination” for golfers from around the country.
What PGA events take place at public courses?
now you’re gonna need to be careful with your use of “PGA.” 
Just off the top of my head, Pebble is public, the Western was played at Cog (now the every-other-year BMW or somesuch), BethPage hosted the US Open a couple of years back. You can play the TPC courses. Quite a few tournament courses can be played if you stay at an associated resort.
I’m not sure of your definition of “destination” but if you mean “vacation destination” then Augusta certainly doesn’t qualify. You must either be a member (good luck, Gerald Ford was turned down at one time) or be a guest of a member who is also thoroughly vetted through the committee.
IOW, it is extremely difficult to play at Augusta. They don’t let vacationers play there. Now Pebble Beach, OTOH, is a public course…
Not to mention that it is only open 3 months a year.
Pebble Beach is not a truly “public” course. You cannot just go to Pebble and play. There are a VERY limited number of “public” tee times reserved, usually late in the day at a time when you will have trouble finishing the course before dark, IIRC.
The rest of the tee times are reserved for guests of the Pebble Beach Lodge. This makes it properly a “resort” course. It is only “public” in the sense that it is not limited to “private” membership only.
Maryland National Golf is locate 5 minutes outside Frederick, Maryland, and is a public course. “National” is just a marketing tool.
Oh, and the PGA Tour plays each year at Torrey Pines Municipal Golf Course, in San Diego. This year, it is also hosting the U.S. Open.