I've written broadly about the origins of some of the weird terms the Klan used in the fraternal traditions which it built off of in this older answer. I would append briefly to that though to touch on the Klan's name itself as I don't address that. It fits within that tradition, and should be understood in that context, but the specific name itself has fairly simple origins.

At the first meeting, probably held in June, 1866, and attended by Frank O. McCord, Richard Reed, John C. Lester, Calvin Jones, John Booker Kennedy, and James Crowe, possible names were discussed. Several possibilities were discussed, before hitting on the term Kuklos, Greek for "circle". This turned to Ku Klux, and 'Klan' was spelled with a K because even diehard racists can't resist the appeal of alliteration. Again, this harks back to the fraternal traditions, and the carnival imagery that the Klan adopted. Even the term itself - 'Circle' - needs to be considered in the context of the Knights of the Golden Circle, a prewar organization which is often seen as a precursor, and were compared to in the Klans early days.

Elaine Frantz Parsons. Ku-Klux: The Birth of the Klan During Reconstruction. UNC Press Books, Nov 2015.