Joe Lane of course being Breckenridge's running mate during the 1860 election. If the 1860 election somehow went to the house, Lane probably had a huge chance of becoming the President. The senate was heavily Democratic, and would of course support one of their own over a Republican.
The House is where the real battle is. Breckenridge and Lincoln are obviously in the top two slots, if there is a deadlock, it is likely because Douglas managed to carry a few free states, so we can give him the third slot if we want (but he doesn't really matter). The congressional delegations are split as such:
Republicans: 15Pro-Douglas Democrats: 1Pro-Breckenridge Democrats: 13Ex-Whigs: 1Maryland/Kentucky were split equally between Breckenridge Dems and Ex-Whigs.
Missouri was equally split between Douglas and Breckenridge Dems.
With Bell not in the running, the Ex-Whigs in TN, MD, and KY will not break for Lincoln unless they want to be lynched when they return home, so it is either Douglas or Breckenridge. Douglas is the obvious choice for the 'compromise candidate,' but with Lane a lock in the Senate, the Breckenridge Dems are not likely to settle for Douglas. So with the House deadlocked, Lane becomes President. From what little I know about him, he was much more of a fire-eater than his running mate, with his political career largely being destroyed in his home state of Oregon with his pro-secession beliefs. So what kind of damage could a more radical pro-slavery man do as President for at least until 1865 (where a Republican is almost assured the Presidency)?.