His Girl Friday is a remake of The Front Page. My Fair Lady and High Society are musical remakes of Pygmalion and The Philadelphia Story, respectively.
What are some other examples?
Heaven Can Wait 1978 was a remake of Here Comes Mr. Jordan (1941)
Then Down to Earth , (2001) with Chris Rock.
The Shop Around the Corner 1940 was remade twice:
In the Good Old Summertime (1949), which stars Judy Garland and Van Johnson.
You’ve Got Mail (1998), with Tom Hanks and Meg Ryan
Not quite, but close.
Bringing Up Baby 1938 very much inspired What’s Up Doc 1972 including many of the same gags.
Spielberg’s Always is a remake of A Guy Named Joe.
Never Say Never Again, of Thunderball.
The Magnificent Seven was a remake, or maybe more accurately an adaptation, of The Seven Samurai.
Zero Hour!
Bedtime Story->Dirty Rotten Scoundrels->The Hustle
Yojimbo > A Fistful of Dollars > Last Man Standing (with Bruce Willis)
Billy, have you ever been in a Turkish prison?
I can’t watch that clip without LMFAO! 
Also with Bruce Willis: The Jackal, a vastly inferior “remake” of The Day of the Jackal.
Do you like gladiator movies? 
Flightplan with Jodie Foster is loosely a remake of The Lady Vanishes. There have been a couple of other remakes of The Lady Vanishes under that title-- there was one in the 70s with Cybill Shepherd and Angela Lansbury; that one tried to make a sort of black comedy out of it, and failed horribly. The other is a British film that is much more serious than the Hitchcock film.
Speaking of Jodie Foster, she made a film called Sommersby that was a remake of The Return of Martin Guerre, albeit, with time and place changed, since it was an American film, and the original was French.
Anna and the King of Siam became the musical The King and I, and much later, the non-musical Anna and the King.
My Favorite Wife became Move over, Darling, and, along with Anna and the King of Siam was just one of a host of Irene Dunne’s films that were remade. (A Guy Named Joe mentioned earlier was hers too.) Additionally, Love Affair became An Affair to Remember.
The Parent Trap was made twice under that name, but it started out as Twice Blessed, starring an actual set of twins.
The Lillian Hellman play with a lesbian rumor, wasn’t made under its original title “The Children’s Hour” until the 1960s (with Audrey Hepburn), but a non-lesbian version called These Three came out in the 30s, with Miriam Hopkins, who has a supporting role in the 60s film.
Wow. I could keep going, but I’m probably getting ninja’d, so I’d better post.
Dashiell Hammet’s novel, The Maltese Falcon, was first filmed in 1931, then remade as Satan Met A Lady (with Bette Davis) in 1936, before John Huston made the classic version with Bogey, Sydney Greenstreet, and Peter Lorre, in 1941.
A comic version, The Black Bird, starring George Segal, was released in 1975, and there are many other versions with the original name out there.
La Femme Nikita was remade as Point of No Return
Oh, and if this counts-- the French film Trois hommes et un couffin, which translates to “three men and a bassinet,” was released in the US under the more euphonic title “Three Men and a Cradle.” It was followed by the uninspired Hollywood film with the uninspired title “Three Men and a Baby.”
Charade was remade as The Truth About Charlie for reasons that, quite possibly, no one can explain.
Might your handle have something to do with this choice?
For that matter, Wait until Dark bears some remarkable similarities to Charade.
Not really.