Well the first question in my mind is does Davis' standing prevent the Democratic Party from spliting in 1860? To do so, Davis has to find a way to either overwhelm Stephen Douglas' political support or become a dark horse/compromise candidate.
Davis in 1860 was serving in the US Senate after having been Franklin Pierce's Secretary of War. He spent much of 1858 ill, however he did do one very interesting thing: in Boston, he gave a speech advocating against secession by the South.
Another interesting fact: Stephen Douglas died in June 1861, exhausted by the events of 1860.
Now, how implausible would it be to wonder one or both of the following: what would Davis have done had he not been ill in 1858? How effectively could Stephen Douglas have sought the presidency had he fallen ill earlier?
The most interesting route, IMHO, is for Davis to avoid his 1858 illness and enter the Democratic convention in 1860 a figure of greater profile. He is for states rights' but for the preservation of the union (because he is an anti-secessionist, not because he beleives in it a la OTL Lincoln; he beleives the South would lose). I wonder if Douglas and Davis might join forces to create a ticket that could carry 2/3 of the Democratic convention, avoiding a Party split. To enable this, Douglas and Davis would have to stop the Minority Report from being adopted as the Democratic Position on slavery: this is what prompted southern Democrats to bolt, since it included no specific protection of slavery. Perhaps, though, we can still have the split convention, but Davis, rather than Breckinridge wins. Davis might have more reason to seek an accomodation with Douglas. If the two can form some kind of alliance and joint ticket, then the Constitutional Union party likely does not form and the Democratic ticket narrowly carries the election of 1860.
In summation, it might not take much of a POD to avoid the electoral mechanics which produced the election of 1860. However, the question is more complicated than simply WI Jefferson is the Democratic nominee.