Donald Trump: Not sure how widespread his fame was in the '80s. Definitely would have been well-known among New York Times readers, but would a small-town Texan have ever heard of him? Definitely at least since The Apprentice debuted in 2004.
Trump was pretty widely known well before "The Apprentice":
I saw the movie "54" (released in 1998 and set around the late 1970s) recently. The club of the same name that the movie revolved around had a special VIP room. Among the guests were Andy Warhol, Grace Kelly, and, yes, Donald Trump.
Carter, in modern times at least, seems to be cited the most as the 'unknown, out of nowhere' winner of a presidential election. He is a rather rare example of someone who was both unknown *and* not regarded as a serious contender to win a major party's nomination.
Then you have the guys who persistently ran for years. There were twenty years between Henry Clay's first and third runs. Had William Jennings Bryan been successfully nominated in 1924 then he'd have a been a nominee well-known to the public for almost thirty years (three previous nominations, a stint as SoS, as well as being a highly charismatic and controversial figure in general).
