We do not personally subscribe to all these reasons but posit them as likely explanations for McCain's conquest of the Cuban vote in Florida, which is a good harbinger for his success there in November:
McCain is for Cuba. It is doubtful that any other Republican candidate had ever spoken the word "Cuba" in a debate or any other occasion in his entire public life except John McCain. This became quite obvious when Romney bestowed Fidel Castro's slogan ("¡Patria o Muerte. Venceremos!") on anti-Castro Cubans and Huckabee revealed that he did not know Cuba was on the State Department's list of terrorist states. McCain, who lived the dark story of Communism in flesh and bone, has never forgotten who his torturers were and has always been a consistent foe of Communist Cuba. His avowal that he would continue the "Wet Foot/Dry Foot" policy because he knows no other alternative (how about the policy that was in place for 30 years before the "WF/DF" policy?) may not weigh as much with older Cubans who don't have relatives on the island or those who regard the newcomers as suspiciously unorthodox in their conduct and tending toward accommodation rather than confrontation with the regime (whether this or true or not).
McCain is an old Cold War warrior. And so, of course, are Cuban exiles. Cubans know that McCain is not a Commie because he was imprisoned and tortured by Commies (and by Cuban Commies, at that). Many of them were, too.
McCain is old. And Cubans are old, too. The median age for Cuban-Americans is twice that for other Hispanics. Besides sharing the same temporal frame of reference, McCain benefitted from the fact that Cuban exiles are wary of young leaders having been monumentally shafted by one 50 years ago. [The Díaz-Balarts are extensions of their father (like George W. was an extension of his father) and Ros-Letihnen is a woman, a well-known quantity and not exactly young anymore].
McCain is not an enemy of paternalistic government. And neither are Cuban-Americans. This is what riles the fundamentalists in the Republican Party, but it is of no concern to Cuban exiles, and may, in fact, be a plus. Cubans don't regard a paternalistic government as a bad thing. This has nothing to do with Castro, whose regime has never concerned itself with the well-being of the Cuban people. It is the Constitution of 1940, abolished by Castro, the most socially-progressive of its time, which is the basis of Cuban support for what has come to be known as the "welfare state."
McCain is not a xenophobe (or at least not a willing one). Cuban exiles know that they are foreigners and it doesn't matter to them and they are shocked that it should matter to anyone else except as a reason to praise them. They are comfortable with who they are and take pride in the fact that they have contributed as much to this country in their post-1959 history here as any other group and more than most.
Although the xenophobes in the GOP have gotten McCain to repudiate the veiled amnesty provisions of McCain-Feingold while hugging for dear life the Wall of Infamy, it is obvious that his sympathies are still with Ronald Reagan's conception of an inclusive America and not Newt Gingrich's "besieged wagon train" mentality. Once president McCain will not be beholden to the xenophobes that despise him regardless of the apparent kow-towing to their anti-Hispanic fixation, and so, Cuban exiles expect, he will not be a danger to them.
The "Contract (on) America" taught Cubans that Republicas like Romney and and Huckabee begin by scapegoating illegal immigrants and end up penalizing the legal ones as well. In the mid 'Nineties, within the living memory of many who had just retired or were approaching retirement age then, Republicans unleashed a panic upon the Cuban elderly when they excluded non-citizens (legal or not) from the social compact and forced them to chose between becoming citizens in their 80s and 90s or being thrown out on the street. (Many chose suicide instead). Whenever the Newt-clones dump on illegal immigrants, Cubans know that they make no distinction between legal or illegal, all foreigners are anathema to them and they are determined to make life impossible for them in this country until they have no choice but to return home . That option, of course, is not open to Cuban exiles, at least not yet.
McCain was regarded as a winner and Cuban don't go for losers. Giuliani had enthralled Cubans in the beginning because he is Latin and knows how to connect with Latins, though his history certainly did not entitle him to much consideration. At the end, though, he had sat out all his early momentum and lost much of his Cuban support to McCain, though retaining enough to beat out Romney and Huckabee among Cuban-Americans voters.