Pierce had little support outside the South after the violence in Kansas. The obvious alternative was Douglas, but (1) northern Democrats, while they realized they would have to defend the Kansas-Nebraska Act, might still be reluctant to nominate someone as closely associated with it as Douglas, and (2) southern Democrats were worried by the association of Douglas with "squatter sovereignty." The Democrats were therefore likely to look for another candidate. Lewis Cass was too old. One possibility was former (and future) governor of New York Horatio Seymour (he took himself out of the running in OTL, but might not have done so if Buchanan was not in the race). Or it could have been someone else--as 1844 and 1852 show, the Democrats had a tendency to nominate "dark horse" candidates in that era.
Because he was such a failure as president, we tend to forget that Buchanan was in some ways the ideal candidate for the Democrats in 1856. He had been out of the country when the Kansas-Nebraska bill was passed; in contrast to the brash Douglas, he had the reputation of being a seasoned, experienced statesman and might therefore get the votes of conservative Old Line Whigs who might otherwise support Fillmore; and he was from the key state of Pennsylvania. It is hard to think of any other candidate with comparable advantages.