Two significant attempts were made in the 20th century to run a regional Southern candidate for president, the Dixiecrats in '48 and George Wallace's American Independent Party in '68. However neither were meant as long term experiments in creating an independent Southern force in politics. That said, however, it is quite plausible that such a force could have eventual come into creation. Chances are, furthermore, with a strain of thinking being developed independent of the ideological mainstream, talks of Southern regionalism, autonomy, or independence would doubtlessly begin and from there a position of independence plausible.
In particular I'd point to the late 19th century as the most likely POD. After the compromise that ended reconstruction, Democratic dominance of the South became total but the situation for still-angry southerners remained static, even with a Democratic president. In such a situation, the emergence of a more populist, more strongly southern independent force might well emerge.
How this will effect politics, both nationally and in the south is more questionable. Likely such a party would be in the minority for years at first, unless a truly massive exodus occurred, especially so long as William Jennings Bryan remained prominent in the party. However, over the years, as the Democratic party moved further to the left, particularly on social issues, they would become an increasingly important force in politics. Yet, on the national level, this may well lead to a more left wing north without the parties having to try to appeal to the south.