The most believable conspiracy theory that I've come across is that it was a hit ordered by Sam Giancana, as outlined by Selwyn Raab in his book, The Five Familes.
The Mafia had the motives and the methods and, until the wiretaps in the '80s, a pretty tight, loyal, and secure organization. But even Raab is careful to say that it is only speculative. Many people across the political spectrum hated LBJ – remember "MacBird"? – and were eager to soil his name and diminish his accomplishments.