In order to avoid secession and civil war, you need to have a POD big enough and early enough to completely alter the the two diametrically opposite paths that the country was on. I don't think that the elimination of say the events surrounding Bleeding Kansas, or the Supreme Court's decision in the Dred Scott case, or electing someone other than Buchanan in 56 or Lincoln in 60 is going to change the country enough to completely avoid secession and/or a war, these events just may delay the inevitable clash of the two regions. The challenge in my mind is to come up with as late of a POD as possible to change the the culture, economics, and beliefs of either or both the North and the South to make them similar enough that secession is not even an issue.
For instance, in the 1830s, cotton wasn't quite the King it would become by the 1850s, and states like Mississippi, Alabama, Louisiana, Arkansas, and Texas were still somewhat sparcely populated. In Virginia, which always was the most populous southern state (by a wide margin), the climate and soils were not really condusive to growing cotton and so slavery there was in a decline. In OTL, many Virginia planters and slaveowners generally either sold off their slaves usually to planters in the Deep South, or migrated with their slaves to these more virgin areas of the Deep South and set up their plantations there. As such, the cycle was continued and slavery continued to become more entrenched and more intertwined in the southern economy. However, if Virginia had just a bit more of a progressive government in the early 1830s, it could have passed a law requiring gradual emancipation of all slaves in the state, with recompense to the slaveowners. While this would not have stopped the sale of slaves down to the Deep South or prevented many Virginians from migrating out of the state, it would have slowed it. Then with Virginia leading the way, states like Maryland, Delaware, and Kentucky might follow over the next few years and emancipate their slaves. At that point the power held by the southern states is broken forever, and it is difficult to see where states like Texas, Arkansas and Florida are admitted to the Union without some type of preconditions. Then over the 1840s and 1850s, while the deep south may still be completely mired in slavery, the region would be isolated from the great majority of the country, especially as other states like Tenn and North Carolina and Missouri emancipate their slaves. At some point, states like South Carolina could be isolated, threatened, etc. and forced (without war or secession) to give up slavery. Eventually, slavery is completely abolished in the country without war or secession.