I'm NOT A HISTORIAN but I've studied this topic a lot and have sources to cite. Here goes:

The term "Illini" was coined in the 1870's. At the time, it was somewhat similar to Hoosiers in Indiana or Sooners in Oklahoma. A word tied to the University and the people of the area. 1/

Because it eventually became Fighting Illini and because Native American imagery was later attached (and especially because the Chief was named "Illiniwek"), most believe the term came from an Illini tribe. This was not the case, at least not specifically. 2/

The terms "Illiniwek" and "Inoca" were used to describe several tribes in the region who spoke similar languages, but by the early 1800's all had consolidated into the Kaskaskia and Peoria tribes. The French version of the word Inoca (Illinois) named the state. 3/

So the timeline looks like this: In 1874, the student newspaper coins the term "Illini" and changes the name of the student newspaper from The Student to The Illini (and, later, Daily Illini). The word "Illini" originates right here in this text (January 1874): 4/

For nearly 50 years, the term Illini exists to describe the University, it's students, faculty, and alumni. In 1874, sports teams aren't even a thing yet.

As those teams come on board in the early 1900's, several terms are used to describe the teams (including "Homecomers"). 5/

The University eventually settled on Fighting Illini and there's debate as to where the fighting came from. Some sources say it was specifically tied to WWI and the "Fighting Illini" overseas, some say it was the push for the stadium in 1921, some say it was used before that. 6/

The 1920's saw the University coalesce around "Fighting Illini" and Native American imagery, as can be seen in this screencap from the Story Of The Stadium (Memorial Stadium) below. There were other pamphlets like this at the time as well (links at the end). 7/

This was then cemented with the introduction of Chief Illiniwek in 1926. The University teams would be known as the Fighting Illini and the Chief would be part of the athletic events. This was the case from 1926 to 2007 when the Chief was retired. 8/

The Chief (and later the war chant) were retired but the name Fighting Illini was kept. Why was it kept? In my opinion, it was kept because the term was introduced by the student newspaper in 1874. "Illini" existed for 50 years before Native American imagery was attached. 9/

The debate now, and the proposed introduction of a physical mascot on the sidelines, IMO centers around whether it's possible to return the term "Illini" to its meaning for the first 52 years or if the 81 years with Native American imagery attached mean that's not possible. 10/

That's how I would frame the senate resolution about keeping Fighting Illini but adding a mascot. It's been 13 years since the Chief was retired, and as the resolution reads, they don't feel there's much distance gained from the Native American imagery. They suggest a mascot. 11/

Sources:
This article from the Library Archives blog has almost all of the info above: https://archives.library.illinois.edu/blog/fighting-illini-name/
The coining of the term in 1874: https://archives.library.illinois.edu/features/illini/illini.pdf
The term "Illini" was not used to describe the Native American people in the region: http://www.museum.state.il.us/muslink/nat_amer/post/htmls/il_id.html 12