Maybe leave the revolution in Cuba, but make sure that Bobby Kennedy doesn't go after them in the 60s. Possibly have an administration that supports them, is tied to them, or is bought by them as they are all forced back into the states. Could even still be the Kennedys only more thoroughly under the thumb of Giancana.
Or under the thumb of Giancana in any way, shape or form. The Kennedy's were no more affiliated with the mafia than any other Democrats...or Republicans. Papa Joe was a stockbroker and had a hand in the cinema; he was more than well enough off, his business was secure and he was making a killing, whereas rum running let alone connections to the mob were dangerous beyond belief. So Papa Joe was in no way connected to the mob, nor were any of the Kennedy clan. The Kennedy brothers had to go through review by members of committees and so forth throughout their career for various reasons. Not once did anything of rum running or mob ties come up. I believe like many myths (especially dirty myths) of politics, this came from electoral campaigns. The slander of a mob-tied Kennedy clan would have been a good smear campaign, and that myth could be tied into charges of a corrupt election (note, the Republicans were dirtier than dirty themselves; the Democrats had Cook County and messed with numbers, note sans mob. The GOP had the rest of Illinois and the numbers coming out of their areas were extremely fishy). It has since been tied into the assassination, and been repeated by a disillusioned America ad infinatum. So it's self perpetuating.
Now that little rant ties into another fact more in line with this topic: the mob were very friendly with many politicians, not to say they were campaign contributors or anything like that, but that if they showed a Senator a good time in a casino or whatever, that Senator would overlook what they didn't want looked at. Cuba is a prime example, where the well to do, the wealthy, and the political elite would be drugged up in a sort of Adult's only Disneyland of sex shows, prostitution, gambling, and whatever else. So the mob used that to their advantage frequently to stay out of the spotlight.
