dude read the DI once in a while they talked about the term "fighting illini" and why we got to keep it (seemed to be very different from what you said, but i think it was a kids article no an AP one)
P.S. i go to a lot more sports stuff here than just basketball but i dont get everyone obsession with the Cheif
Edit: Because i find more source
The earliest recorded usage of the term "Illini" appears to have been in January 1874, when the weekly student newspaper changed its name from The Student to The Illini. An editorial (pdf, 150KB) in the first issue of the renamed journal (Volume 3, Issue 1) implies that the term was coined and had not formally existed prior to 1874. During the late 19th century and the first years of the 20th century, it was often used to refer to the students, faculty, staff, and alumni of the University, as well as to the campus as a whole.
and
Application of the term specifically in reference to athletic teams seems to have been first by secondhand accounts of University athletic teams. The earliest reference in the Illio appears to be one mention in the summary of the 1907 football season. The term gained greater frequency in the next decade, especially during the 1914, 1915, and 1916 football seasons. Daily Illini articles and football programs prior to these dates do not extensively cite the term, at least to the extent of our staff's knowledge. Other terms, such as the "Indians," "our men," "Orange and Blue," and the "homecomers" were sometimes applied to the team, none in a consistent fashion, except for "Illinois" and "Varsity".
etc. from our very own school
http://web.library.uiuc.edu/ahx/illini/