Cohn Is the Head
Initially, the head of Columbia Pictures, Harry Cohn, had a lot of ideas for casting From Here to Eternity—and most of them didn't work out. He wanted Humphrey Bogart to play Warden and Joan Crawford to play Karen Holmes. Neither happened—Crawford was nixed after she insisted on being filmed by her own cameraman. Lancaster and Deborah Kerr took the roles, respectively, with Kerr playing against type, since she typically played English duchesses and things like that.
Cohn wanted Aldo Ray to play Prewitt, since he was cheaper than Clift (who was what today might be called buzz-worthy—the young superstar of the moment). But the director, Zinnemann, threatened to resign, and managed to ensure Clift got the role. (Source) .
Also, Cohn and Zinnemann originally wanted Eli Wallach (the guy who played "The Ugly" in The Good, the Bad, and the Ugly) to play Maggio—but Wallach turned down the role to do a Broadway play instead.
The role ended up going to Frank Sinatra, who was down on his luck and badly in need of a break. The Godfather (the book and the movie) later implied that Sinatra got the part by having mafia associates put a severed horse head in Cohn's bed to scare him. However, this simply isn't true (even though it's such a good story). Sinatra probably got the role due to his then-wife Ava Gardner's connections. (Source)
A Jones for Thwarted Love Stories
Another producer was Buddy Adler, who went on to produce other classic melodramas like Love is a Many-Splendored Thing (1955) and Heaven Knows, Mr. Allison (1957). Interestingly, like From Here to Eternity, both are love stories set in far-flung locations: the former in Hong Kong during China's Communist Revolution, and the latter on a South Pacific Island during WWII.
Also, as with Warden's love for Karen Holmes, both films deal with love that wouldn't be officially approved by society: the main character in Love Is a Many-Splendored Thing has an affair with a half-Chinese nurse, earning the condemnation of conventional Hong Kong, while the main character in Heaven Knows, Mr. Allison falls in love—with a nun. (Nothing happens, though).