What would it take for John Breckenridge to be eligible for President of the Confederate States? IIRC, the convention elected Jefferson Davis on 18 Feb 1861, so it'd have to be before then.
Is there a plausible way to get Kentucky to secede before then? Maybe a stronger abolitionist sentiment from the North?
Or could it be as simple as Breckenridge moving to MS in January of 1861? Did the CS Constitution have laws regarding how long one had to live in a particular state? If so, would the convention follow such laws if Breckenridge had moved? (I pick MS because Breckenridge won 60% of the vote in the 1860 election, a larger percentage than any of the other Southern states.)