Actualy, I think it's because it's in Normal, IL.
And Normal, IL was named as a city after the fact that they had the Illinois State Normal Schools.
"Normal School" is a term that is used to describe people being trained after they finished their elementary education, to be trained as public school teachers, as described in this link:
THE NORMAL SCHOOL
On July 3, 1839, three young women reported to Lexington, Massachusetts, with hopes of attending the first state funded school specifically established for public teacher education (what were then referred to as "normal" schools). After taking an examination which determined they were satisfactorily versed in the subjects taught by the ordinary district school, they were granted admission to this experimental program, the first in the nation.
Here's the background on the history of Illinois State University, or what was originally called Illinois State Normal University:
Illinois State University was founded in 1857 as the first public university in Illinois. The institution has a long and distinguished history. Among those influential in its establishment were Jesse Fell, who organized the campaign to locate the institution north of Bloomington, and Abraham Lincoln, who drew up the bond guaranteeing financial commitments to the new institution. The university was originally established as a teacher education institution with the name Illinois State Normal University. In 1964 the name was officially changed to Illinois State University, reflecting a broadening of the mission and offerings of the institution.
And here is a quote from the following website, that shows the original name of Normal, IL was North Bloomington, IL, and then they changed it after the city was chosen as the site of the Illinois State Normal University:
The History of Normal
In 1854 the town of North Bloomington was platted in the area which was commonly known as "The Junction," which was located at the intersection of the Illinois Central and the Chicago & Alton railroads. ...
In 1857, Governor William Bissell signed a bill to create a normal school. The term "normal" was based on the French teaching schools and was the general name for all schools set up to be teachers’ colleges. The bill stipulated that the permanent location would be the place that offered the most favorable inducement. Jesse Fell took up the campaign for Bloomington and obtained financial backing totaling $141,000, which surpassed the closest contender of Peoria, which raised $80,000. Abraham Lincoln, in his capacity as an attorney, drew up the bond guaranteeing that Bloomington citizens would fulfill their financial commitments. The University first held classes in Bloomington while the campus was being built north of Bloomington. Old Main, the all-purpose building for the University, was completed in 1861 and the state’s first public institution of higher education had a permanent home.
In February of 1865, the Town was officially incorporated under the name of Normal. Under the formal State of Illinois charter, a town government of five trustees was elected, sidewalks were constructed and the sale of "intoxicating drinks" was prohibited. This prohibition remained until the early 1970’s.
I see that you are probably a North Dakota State University fan, by your user name. So, in the state of North Dakota, most likely schools such as Minot State University and thingyinson State University were most likely estasblished as "Normal" schools. Every state has at least 1, and many time, multiple regional "Normal" schools. These are usually the "directional" named schools, such as Northern Iowa, Northern Illinois, Western Illinois, Southern Illinois, Eastern Illinois. I believe Minnesota State University-Mankato, Minnesota State University-Moorhead, Bemidji State University, and St. Cloud State University (all in Minnesota) were originally "Normal" schools, just like Illinois State University.