First of all, I'd like to correct you. Presidents and Vice-Presidents were elected to 1-time 6-year terms, though I think the Vice-President could take office again (just not the President). Search "six years", and you'll find it. That's why there wasn't another election that Davis had to go through during the war; it had nothing to do with the war going on, that just wasn't how it was set up.

If John Tyler could've gotten past most of his general faux-pas during his time as president of the United States, he likely could've been a candidate. He did support the Confederacy, and was a member of the Confederate House of Congress for Virginia.

Former Presidents Fillmore or Pierce, who were both supporters of slavery (if not the Confederacy, not sure on that count), could've conceivably been elected.

However, had the Confederacy won independence (which would've been necessary, for them to elect another president), there would've been a few more candidates.

One of the most popular (obviously) would've been Robert E. Lee. Similar to Grant's election in the United States after Lincoln's death, Lee was very popular for his military prowess in the South, and would've had huge prospects for a future political career (if he chose to pursue it). Other than that, Davis' vice-president could've conceivably run (Alexander Stephens). But those are just my best guesses, maybe someone with more knowledge on Southern notable figures could provide you with more information.