Leo McCarey remade his own Love Affair (1939) with An Affair to Remember (1957). I like both films a great deal. I think that McCarey was at his best when he showed charity in the form of tact and quiet self-sacrifice. Anyway, in each film he had an actor-actress pair that was brilliant, with actresses who knew how to UNDER-state things: Charles Boyer and Irene Dunne in the original, and Cary Grant and Deborah Kerr in the remake. Dunne and Kerr are in my all time top 10 for Hollywood actresses, but neither one of them ever won an Oscar. Throw in Barbara Stanwyck for the trifecta ... On the male side, how the heck Edward G. Robinson, Claude Rains, and Cary Grant never won an Oscar, I have no idea.
William Wyler worked on the original silent-film Ben-Hur, and then did his own remake many years later, with Charlton Heston, Stephen Boyd, Jack Hawkins, Haya Harareet, Sam Jaffe, Martha Scott, Finlay Currie, and the flat-out scene-stealing Welshman, Hugh Griffiths, playing the Arab sheik. The movies are sufficiently distinct, so that you don't end up comparing Raymond Novarro with Heston -- a very different kind of Judah Ben-Hur. The supporting cast is very strong -- that oily bass sort-of British Irish accent that Stephen Boyd uses is perfect. A lot of Heston's movies rely too much on him; this one doesn't.
And then there's Titanic ... After the tense and no-misstep A Night to Remember, with Kenneth More (perfect, and understated; you don't need to exaggerate the Titanic disaster), somebody comes up with that expensive dog...
Frank Capra re-did his Lady for a Day (1933) in Pocketful of Miracles ... that was a mistake. Andrew McLaglen did a sort-of remake of John Ford's The Quiet Man (in which his father Victor and John Wayne get into the longest and merriest fist-fight in Hollywood history) in McLintock -- and though McLintock is fun to watch, it's a lot of rehashing, and mainly a vehicle for getting Wayne and Maureen O'Hara back together (and Maureen O'Hara never won an Oscar either; go figure).
If a movie is worth remaking, it is also perilous to try, because your audience always has the original in mind, for comparison. Imagine somebody being so dumb as to try to remake The Sound of Music. Who's going to stand up against Julie Andrews, for singing?