Safire's Political Dictionary by William Safire says this under initials, presidential.
A contribution made by newspaper-headline writers to American history. Before 1932, the President was referred to in headlines as the President, or by his last name, occasionally by his nickname ("Teddy," "Cal"), once by his initials (T.R. for Theodore Roosevelt). Presidents, in signing brief memos, would often use the initial of their first name (A. Lincoln).
Franklin D. Roosevelt initialed memos "FDR" and his staff referred to him that way. This proved a boon to headline writers, saving six spaces on both "Roosevelt" and "President." Use of initials was terse but not disrespectful, as "Frank" might have been. English newspapers had long followed the practice: Gladstone was headlined as "GOM" (Grand Old Man). Following FDR, HST was immediately adopted as short for Harry S Truman. Though Eisenhower initialed short notes "DE," his nickname was short and famous enough to take the place of initials, though "Ike" was considered too familiar for The New York Times.
Irreverent headlines writers who referred to John F. Kennedy as "Jack" before his election quickly switched to "JFK" afterward. Ted Sorensen wrote: "JFK - as he persuaded the headline writers to call him, not to imitate FDR but to avoid the youthful 'Jack.'"
Even before his association with FDR [I think he means JFK], LBJ was preoccupied with initials: Lady Bird Johnson, Lynda Bird and Luci Baines all carried the same initials, as did his ranch [and his dog Little Beagle Johnson]. "All the Way With LBJ" was used as a preconvention slogan in 1960, but achieved little national recognition. Johnson became LBJ outside his own circle only after he became President. In 1964 he used his initials in campaign advertising: "USA for LBJ."
Jimmy Carter avoided using his initials because "J.C." is usually associated with Jesus Christ. Richard Nixon's last name was short, which meant headline writers did not need "RN"; miffed, the former President used initials as the title of his 1978 memoirs. Jerry Ford's name was even shorter, doing away with the need for a gruff "GRF." Neither Reagan nor the elder Bush used their initials widely, but initialese cropped up with the Clinton term as FOB--not "freight on board," but "Friends of Bill." Headline writers seized on the middle inital of the younger Bush's name, both because it differentiated him from his father and because the irreverent, drawling "Dubya" came with him from Texas.
There is also some interesting discussion of this question at: The Straight Dope