Lake Illinois is what the French originally called Lake Michigan, naming it after the Illinois Confederation (Illiniwek) of tribes that controlled the area. It was an important geographical landmark referred to in many early histories of North America; at this time, Lake Michigan/Illinois was considered a separate lake from Lake Huron. Lake Illinois was, for a time (c. 1700), the Southern border of New France, although they would not take long to expand their territory down to the Mississippi River to the Gulf of Mexico, before surrendering the territory to Great Britain and Spain.
Some time in the mid-1800s the use of Lake Illinois finally died out, and Lake Michigan, long the favorite, was the name both officially and in popular usage. Michigan is indeed the more appropriate name, being the name not of some local governing body, but coming from the Ojibwe word michi-gami meaning "great water".