Honestly, one of the primary reasons to use a middle initial in your political life is branding. In some cases, it's a necessity, like George W. Bush's middle initial, which differentiates him from his father. But generally politicians can find their given names unwieldy or awkward and thus decide to add a middle initial in the same way an actor might, to make themselves sound more appealing. John F. Kennedy used his to avoid the youthful 'Jack', which he went by with friends and family, being used in the press. Being the youngest elected president, he was eager to project the maturity that 'JFK' and 'John F. Kennedy' have but 'Jack' does not.

In post-FDR America several political leaders began to brand themselves with their middle initial, and by extension a three-initial moniker, some to greater success and some to less, in order to tap into Roosevelt's popularity. FDR, incidentally, was taking after his cousin Theodore who was well known as TR, because 'Roosevelt' was unwieldy for newspaper headlines.

For example, Lyndon Johnson was relentless in the development of the LBJ brand, initially because in his first Congressional campaign he tied himself to Franklin Roosevelt as closely as possible. He wore a belt buckle styled in the shape of an LBJ, he made sure press releases named him as LBJ, and he had a dog named Little Beagle Johnson. And that's leaving out the fact that both of his daughters' initials were LBJ and his wife Claudia Johnson went by Lady Bird.

Richard Milhous Nixon also tried desperately to be referred to by his initials RMN, partially as a show of power and also, I suspect, as a representation of his feelings towards the Kennedy family, JFK and RFK obviously being beloved figures by the time he became president. For whatever reason, perhaps the total un-catchiness of RMN or the obvious forcing of it as a reaction to his predecessors, it never caught on. He eventually dropped the M when releasing his memoir RN but that didn't work either.

Sources: Caro's The Years of Lyndon Johnson and Evan Thomas' Being Nixon.