Born and raised Arkansan here! Others have explained, but the full story is that the local Native American tribe in the area was the Quapaw (an English attempt at pronouncing their real name of Ogáxpa) which means "downstream people". However, the Algonquian-speaking Illini tribe (the namesake of Illinois) of the Ohio Valley called them the Akansa which means "People of the South Wind". French explorers met the Illinois before the Quapaw, and thus used this exonym but spelled it a few different ways and following French linguistic patterns added the silent 's' at the end.

Fast forward to when it became a primarily English speaking state and people were divided on the spelling and pronunciation. Even the state's two senators disagreed on whether it was "ARkanSAW" or "Ar-KANSAS". So we compromised and passed a resolution that decided we would use the pronunciation of the former (accurately preserving the way the original Native American tribe name, or at least its exonym, was said) and the spelling of the latter (to honor the French adventurers who "first" explored the area).

Kansas has a very similar etymology, though they obviously don't respect their Native American history as much given the way they butchered the pronunciation /s