Several sources give conflicting answers to this question. The most common explanation comes from "The Ozarks: The American Wilderness": 

One theory suggests that “Ozarks” came from the French name for the Osage Orange Tree, or Bois d’Arc.

Another theory suggests the word is a corruption of “aux arcs”, meaning “with bows”, and used by French trappers to describe the local Quapaw Indians.

Yet another theory, and considered by some to be the most accurate, states that French Trappers shortened the phrase “aux Arkansas”, meaning “going toward Arkansas”, to “aux Arks”. The word “Arkansas” refers to Arkansas Post on the Mississippi River. Arkansas Post is today a National Memorial under the National Park Service. In addition, Morris Arnold’s "Colonial Arkansas 1686-1804: A Social and Cultural History" is an excellent history of Arkansas Post.

An English traveler, John Bradbury, first used the name “Ozark” in print in 1809, and the term “Ozark Mountains” first appeared on a map made by S.H. Long in 1815, thus “officializing” the name, according to Phyllis Rossiter’s "A Living History of the Ozarks".