It seems I get this question about 3 or 4 times a month from visitors to the area. Well, to answer that question, I have found an old article from 1932.
Confronted with many questions of varied nature the Jefferson city Chamber of Commerce which has become adept at answering queries, almost met a Waterloo Thursday when an elderly couple, en route home from a vacation trip at, the Lake of the Ozarks, stopped here and asked what the meaning of “Ozarks" happened to be. After a search, which for a while proved discouraging, the chamber found a report by Miss Jeffries in a Lake of Ozarks publication which explained that, the derivation of Ozarks was from the French "aux arcx," which means "with bows and archery", and not from the Indian as has been popularly supposed. "When the explorations of the French carried them into this j section, the only known inhabitants were Qaw Paw Indians, a branch of the Sioux. They migrated here from the Ohio and Illinois rivers when the white man came," Miss Jeffries said in her explanation. "The French established the first trading post, Aux Arcx in this bow and arrow country. Later the Spanish came questing for hidden gold. Many of their hidden treasures, repose no doubt, beneath the Lake waters of the new inland sea." The local Chamber of Commerce invites anyone having a different explanation for "Ozarks", to write and tell them about it.
“Ozarks is from the French, The Bows and Archery.” Jefferson City Post-Tribune 66.183 (15 July, 1932) A1-1. Access Newspaper Archive. Donald W. Reynolds Library, Mountain Home, AR. 1 Oct. 2010.