The standard scenario here is Clay winning in 1844. See Gary Kornblith's "Rethinking the Coming of the Civil War: A Counterfactual Exercise"
http://www.oah.org/site/assets/documents/02_JAH_2003_kornblith.pdf
"The key to peace in my counterfactual scenario is a victory by Henry Clay over James K. Polk in the very close presidential election of 1844. Had Clay won 5,107 more votes in New York State (out of more than 485,000 cast there), he would have become president. As a result Texas would almost certainly have remained an independent republic in 1845, and the United States would not have gone to war with Mexico the following year. Based as it is on a highly plausible turn of events, this counterfactual scenario promises to advance our understanding of thecauses of the Civil War...
"Having established that a Clay victor y in 1844 was thoroughly plausible, we can, proceeding with our thought experiment, consider the implications of such a victory for American political and social development. First, we must ponder the fate of Texas. Unable to point to the electoral outcome as an endorsement of his policies, Tyler would not have succeeded in pushing thro ugh Congress a joint resolution authorizing annexation before he left office. At the urging of the president-elect, Whigs in the Senate—southern as well as northern—would have cited Clay’s victory to justify their continued opposition to annexation. Had Henry Clay taken office as president on March 4, 1845, he would have enjoyed a good deal of flexibility in crafting his policy toward the Lone Star Republic...
"In 1843 George Gordon, Lord Aberdeen, the British foreign secretary, floated a proposal coupling Mexican recognition of Texan independence with abolition and British assistance. Sam Houston, then president of the Texas republic, briefly displayed some interest. But Aberdeen withdrew the suggestion of a quid pro quo the following year, and thereafter he temporized on the question of Texan abolition for fear of encouraging American annexation. Given competing foreign policy priorities, th e British were unprepared to risk major involvement in Texas in the face of American opposition. A President C lay would have objected to any British effort to promote abolition in Texas for the same reason he opposed annexation of Texas: his overriding concern was the maintenance of sectional harmony and American political stability.
"In all likelihood, Aberdeen and Clay would have joined diplomatic forces in support of Texan sovereignty. In early 1845 the B itish and French undertook a new initiative to convince Mexico to recognize Texan independence , and Mexican authorities reluctantly agreed. Although the Mexican government would have felt less compelled to comply had Clay rather than Polk been elected, it could not have comfortably ignored the combined pressure of Great Britain, France , and the United States. We may postulate that sooner or later during Clay’s presidency Mexico would have recognized Texan independence and entered into international arbitration over boundary issues. Even had the Mexican government continued to refuse official recognition, it would probably have shrunk from open war fare and allowed the Lone Star Republic to consolidate authority and power further..."
Kornblith argues that with Clay as president and no Mexican War, the parties would continue to divide on economic issues (banks, tariffs, etc.) with the slavery issue muted. The whole Wilmot Proviso controversy of OTL would not take place, and it was this controversy which gave birth to the Compromise of 1850 with the application of "popular sovereignty" to the newly acquired southwestern territories. It was this same doctrine of "popular sovereignty" that was applied to Kansas-Nebraska in 1854, with fatal results to the second party system, and the emergence of a new party, the Republicans, whose victory would lead to southern secession...
(There are all sorts of problems with this scenario, as Kornblith recognizes. For one thing, the expansionist Democrats, having lost in 1844, might win in 1848...)