Nineteenth-century American politics? Excellent! There's really two answers to that question which come to mind:

Firstly, there's the possibility that this is just boilerplate. In antebellum political discourse the office sought the man, not the other way round. To maintain the fiction that nobody sought major political office, candidates did not campaign personally until Stephen Douglas went South in 1860 and parties discussed their candidates' chances and campaigns obliquely in terms of 'availability' rather than openly discussing whether their candidate had a chance of having their dreams of officeholding fulfilled. So it's possible that Buchanan just played to this antebellum discourse to appear like a reluctant statesman would would relinquish power.

Secondly, there's the possibility that he genuinely only wanted one term in office. In today's two-term system - which didn't exist in 1856 - this might sound surprising. But in the 1840s and 1850s, the idea of a one-term presidency was quite common in Whig political circles at the very least, and it featured in many of their platforms in this period. Buchanan, though, wasn't a Whig; he had a long career in the Democratic Party behind him. It's possible that he wanted only once shot at the Presidency because of his age after that long career, but my gut feeling here is that someone who thrived on public office may well not have wanted to limit himself this way. The alternative is that in 1856 the faction-ridden Democrats turned to the respected Buchanan for one term as a safe option - he wouldn't have any incentive to use federal patronage to reward supporters and punish Pierce and his supporters. This could have been a way to refuse Pierce's nomination (as he was unelectable) without threatening his supporters and the incumbent patronage army. If so, events in Kansas made sure this plan failed.

If anyone has done to the Democrats what Michael Holt did for the Whigs in The Rise and Fall of the American Whig Party, that would be the place to go, but I can't think of anything offhand. So while I can't say for certain, to me it makes sense that Buchanan would seek one one term because in 1856 that kind of aspiration didn't invite public ridicule (quite the reverse - it fitted in with contemporary political discourse) and it was what Democratic politicians thought was in the best interests of the party at the time.

Also, it's worth thinking about 'greatness' in 1856-7. When Buchanan assumed office, the country was still dealing with the repeal of the Missouri Compromise and violence in Kansas. With the Supreme Court ready to hand down the Dred Scott decision, Buchanan would have believed himself to have a trump card in settling the slavery dispute: a settlement sanctioned by the Supreme Court that abolitionists and fire-eaters would have to obey. Forcing a Supreme Court ruling on slavery in the territories to end all further political debate on slavery had been an intention of Congress for some time. Buchanan's call for respect for the upcoming decision in his inauguration address (even though he knew what the ruling would be) fulfilled his likely definition of greatness rather than anything else he could achieve in eight years. Events obviously proved his and Taney's plan wrong.