On what basis did the south claim the right to secede?

After the election of 1860, several southern states seceded from the Union. While this was clearly an act of rebellion by the southern states, the South believed they had a right to secede. In the Declaration of Independence, Thomas Jefferson wrote that governments get their power from the consent of the governed. Jefferson stated that when a government fails to protect the rights of the people, the people must replace that government. The southerners believed that the United States government was no longer protecting their rights. They believed that President Lincoln was going to abolish slavery, even though he said he didn’t plan to do this. The slaves were considered to be property, and the Constitution protects personal property. Thus, the southerners believed that they needed to get rid of the old government and replace it with a new government.
http://www.thelibertypapers.org/2007/01/14/did-the-south-have-the-right-to-secede/

https://theimaginativeconservative.org/2012/12/the-right-to-secede.html