The challenge here is managing to get the election to the point where no one has a majority in the Electoral College, and thus throwing it to Congress. I myself have not been able to do that, as the minimum of electoral votes for victory is 152, and swapping the states that were closest between Lincoln and Douglas to Douglas still gives Lincoln a majority. From there, the what if comes into play of the presidential/vice presidential decisions and its aftermath.
At the 1860 RNC:
Day 1: opening ceremonies.
Day 2: platform presentation. Someone proposed proceeding to nominations, but the tally sheets weren't ready, and the delegates preferred to adjourn, rather than wait for the sheets to arrive from the printers.
Night of Day 2-Day 3: Lincoln's managers positioned him as the leading opponent to Seward, and made the deal with Cameron for Pennsylvania's votes.
Day 3 - nominations.
PoD: tally sheets are available on Day 2. Seward is nominated.
Effects: Seward was well-known as a "radical" on the slavery issue. Lincoln was "sound", but not "radical". Seward also had some baggage on
immigration and religion: as governor of NY, he'd supported state aid to Catholic parochial schools, which was poison to ex-Know-nothings, who had nearly all turned Republican. Some of them would bolt to ex-Whig John Bell, the "Constitutional Union" candidate; others would because of Seward's "radicalism", or switch to Douglas.
Douglas would also have a native-son advantage in Illinois, and in Indiana against easterner Seward.
So: Douglas carries Indiana, Illinois and California; Breckinridge carries Oregon. Seward gets 149 EV, 3 short of a majority. As a side effect: Most of the 6,000 votes that Lincoln got in Maryland go to Bell instead of Seward, and Bell carries Maryland.
Final score: Seward 149, Breckinridge 67, Bell 47, Douglas 40. The House can choose Seward, Breckinridge, or Bell. The Senate chooses the VP, and can choose either the Republican or Breckinridge's running mate, Sen. Joe Lane of Oregon.
Note that it would be the current Congress, elected in 1858-1859 that chooses. The new Congress would not take office till March 4, when the new Presidential term also starts.
Republicans control 15 House delegations. Douglas Democrats control Illinois (5-4). Whigoid "Opposition" Representatives control or split several slave state delegations; the rest, and California and Oregon, are controlled by Breckinridge Democrats.
Democrats control the Senate (which votes individually). The Senate will elect Joe Lane.
The House could stall. Breckinridge needs all 15 slave states plus California and Oregon. The one Oregon Representative was an ally of Lane; if he abstains, the House doesn't elect, and VP Lane steps into the vacant Presidency. (OTL, Republicans pointed this out in the campaign, asserting that "Lincoln or Lane" was the real choice.)
Or, Seward might get one of the Douglas men in Illinois, and then buy off either the Oregon or Delaware Representatives. (Both were pro-Southern, but for a high enough price... and for the Presidency the offer could be really good...)
Or, Republicans could diddle with the electoral votes to change the order of finish. For instance, if New York's 35 electors voted for Bell's running mate Edward Everett, he'd finish second and Lane would be out. (The Republican would still be first.)
Another point is that Douglas stated that he would not accept election by the House of Representatives; I don't know why, or whether he would stick to the pledge.