My POD is the "Black Warrior affair" leading to war with Spain in 1854.
http://www.infoplease.com/ce6/history/A0807810.html It might have done so--there were calls for the suspension of the neutrality act, which would mean unleashing filibusters on Cuba--except that the Kansas-Nebraska Act was pending in Congress, and anti-Nebraska forces raised a violent outcry that the administration was looking for war as a way out of its sectional troubles.
The turning point was probably May 30, 1854. On that day, Senators Mason, Douglas, and Slidell--in short, the Democratic majority on the foreign relations committee--met with President Pierce and urged him to support legislation calling for a suspension of the neutrality act. Instead of doing so, Pierce proposed to his callers "the creation of a three-man commission to go to Madrid to present to the government in all seriousness the desire for Cuba and to warn that probably only cession would stop the filibusters. The three visitors accepted this plan, though far from eagerly. As a part of the arrangement, [Secretary of State William] Marcy was called upon to telegraph to the district attorney in New Orleans that decisive measures were on the way. This was to help him hold the filibusters in line. Pierce also promised that before the session ended he would explicitly ask for a big appropriation, big enough for war purposes, in case the commission was unsuccessful. On May 31, i.e., the next day, Pierce issued a proclamation calling for an observance of the neutrality laws." Ivor Debenham Spencer, *The Victor and the Spoils: A Life of William L. Marcy* (Providence, RI: Brown University Press 1959), p. 323.
The result of Pierce's decision was to kill off the filibuster movement. Its leaders, including Mississippi's ex-governor John Quitman, were even required to give bond for their good conduct. Another result was a more conciliatory attitude toward the Black Warrior incident. By midsummer, as it turned out, Pierce had not dared to send Congress the proposal for the commission, though that body was still in session; and the Senate foreign relations committee decided not to ask for an emergency appropriation, though Pierce had indicated his willingness to do so.
This does not by any means indicate that Pierce had given up on Cuba. Something like the originally-planned commission was eventually created and issued the famous "Ostend Manifesto"
http://xroads.virginia.edu/~HYPER/HNS/Ostend/ostend.html but by that time the Democrats had suffered drastic defeats in elections in the North--due largely to a backlash against the Kansas-Nebraska Act--and even Pierce (let alone the more conservative Marcy) had to repudiate the Manifesto.
So basically my POD for US acquisition of Cuba is *no Kansas-Nebraska Act*. Without this, Pierce and the overwhelmingly Democratic Congress would probably have approved a quick suspension of the neutrality act after the Black Warrior affair. And as I stated in a post a few years ago, organization of Nebraska without repeal of the Missouri Compromise was by no means inconceivable. For a while, even David Rice Atchison, despairing of getting repeal through Congress, was willing to accept this, but when other southerners showed an unwillingness to organize the territory on this basis (giving, among other reasons, their well-known respect for Indian land titles
) and when his bitter enemy Thomas Hart Benton started to mock him for his retreat, he swore that he would see the territory "sink in Hell" before giving it to the free-soilers. If just a few Upper South senators had gone along with Atchison's temporary retreat, there would have been no Kansas-Nebraska Act as we know it. There might still be a controversy over slavery in Kansas--the Missourians there might still try to establish it, arguing the Missouri Compromise was unconstitutional, and a Dred Scott-like test case would make its way to the Supreme Court--but at least the political explosion of 1854 would be delayed. (Of course another way to have the Kansas-Nebraska bill as we know it not come up is to have the Black Warrior affair happen a few months before it did in OTL--in short, have the US get to the brink of war with Spain *before* the Kansas-Nebraska bill is introduced. The war scare would doubtless delay any decision about what to do about Nebraska.)
Secretary of State Marcy, never a great enthusiast for Cuba (and especially opposed to taking it by force) pretty much summed up the situation in a letter to Senator Mason on July 23, 1854:
"To tell you an unwelcome truth, the Nebraska question has sadly shattered our party in all the free states and deprived it of that strength which was needed and could have been much more profitably used for the acquisition of Cuba." Quoted in Spencer, *The Victor and the Spoils*, p. 324
The South in 1854 was strong enough to get Cuba--or to get the Missouri Compromise repealed in a futile effort to make Kansas a slave state. She was not strong enough to get both, and disastrously chose the Kansas shadow over the Cuban substance. (Of course the real disaster of Kansas for the South was that it led to the rise of the Republican Party. I doubt very much that a war with Spain, provoked by the Black Warrior incident, would be enough to do so, even if it led to the acquisition of Cuba as a slave state. Unlike Kansas, Cuba already had slavery, so slavery would not be extended by its acquisition; it was even argued that acquisition of Cuba would help stem the illegal African slave trade to that island. And in any event, unlike Kansas, Cuba was not a place where northern farmers were planning to settle.)
