One theory is that the name is a derivation of the Ozark tribe's name for the Osage people, the Wa Zhazhe. The Wa Zhazhe were part of the Dhegiha Siouan groups who called the Osages "Inzashzha," which means "people who live in little tents" or "tent dwellers."
Another theory is that the Osage tribe were named after the river whose name means "clear water." The banks of the Osage River are covered in Osage orange trees. The French trappers who settled in North America named it the Osage orange tree, as well as the Osage River and later the Osage tribe after it.