Here's how "Illiniwek" started. This info is from one old fan long ago, so its hardly official.
The word "Illiniwek" as originally coined is probably pretty close to what we mean when we say "the human race". Originally a few tribes in present-day Illinois used the term to refer to themselves and all the neighboring tribes of which they were aware at the time; so at that time, this was a word meaning "everyone on earth". These few tribes were generally cooperative-no longstanding wars as you might see among other Indians in other territories.
Later, as these tribes found other Native Americans as well as white settlers from Europe, the meaning of "Illiniwek" morphed into what the way it is now commonly used: a word describing a loose confederation of perhaps five tribes. This is how it was translated by the early French explorers. There never was an individual tribe of Indians called "Illiniwek" or "Illini": this is totally dissimilar to the tribes of Navajos, Seminoles, etc.
Today, of those peoples who identified themselves as part of the Illiniwek confederation perhaps the only identifiable descendents left are the Peoria Tribe. They are headquartered in Oklahoma; they are recognized by the Federal government in the same way as every other Native American tribe. Since the time of the French explorers, some of the other Illiniwek tribes were consolidated by treaty into the Peoria or other tribes; others drifted apart over the years and their members either joined other tribes on their own or assimilated in some other way. So the Illiniwek are technically not extinct-but then again, a tribe named Illiniwek never existed in the first place. This certainly complicates any effort to get NCAA-okayed permission: we'd essentially be asking for permission to use a word of their language, if you will.
Nice that the NCAA only chooses to protect those people who originally settled within the borders of 20th Century America, isn't it? I wonder if NAFTA applies? And if the NCAA admits members from Canada, what's to stop them from admitting members from Mexico? Will the Aztecs be sanctioned at that point?