One exception to the predominant US preference for "University of [State Name]" for the most prestigious public university in a state is Indiana University (not "the University of Indiana"). It isn't clear why the state of Indiana chose this wording for its flagship public university, but the name goes back more than 180 years. From Theophilus Wylie, Indiana, University, Its History from 1820, when Founded, to 1890 (1890):
As soon as the four years prescribed by the Constitution of 1816 had expired, the Legislature of the young State made haste to comply with its requirements, and a "State Seminary" was founded, which, through the "Indiana College," ultimately became the "Indiana University."
We have now [in 1838] reached a point of import to the institution, for the "Indiana College" was converted into "The Indiana University" by "an act to establish a university in the State of Indiana," approved February 15, 1838 (Local Laws, 1838, p. 294), and a new era in its history was entered upon. The act is as follows:
"Section 1. Be if enacted by the General Assembly of the State of Indiana, That there shall be, and hereby is created and established a University adjacent to the town of Bloomington, in the county of Monroe, for the education of youth in the American, learned and foreign languages, the useful arts, sciences (including law and medicine) and literature, to be known by the name and style of the Indiana University, and to be governed and directed as hereinafter directed.
Notwithstanding the formal designation of "the Indiana University" as the university's name, the form "University of Indiana" appears nine times in the course of Wylie's 1890 history. Those occurrences suggest a limited degree of informal interchangeability of the wordings at that date. They may also indicate that, even in Indiana, people sometimes slipped into use of the normal US preference for the form "University of [State Name]," despite official endorsement of the exceptional form "[State Name] University."
One indication of the "prestige" element of the wording "University of [State Name]" is evident in the resistance of certain state universities to application of the "University of [State Name]" designation to another university in the same state. The most extreme example of this that I'm aware of involves Texas A&M University, where many students insist on referring to the University of Texas (at Austin) as "Texas University"; the wording even appears—in lowercase letters—in the official Texas A&M "war hymn" ((what at other schools would be called a "fight song"), which dates to 1918:
Good-bye to texas university / So long to the orange and the white / Good luck to dear old Texas Aggies / They are the boys that show the real old fight / “The eyes of Texas are upon you ...” / That is the song they sing so well / So good-bye to texas university / We’re going to beat you all to Chig-gar-roo-gar-rem / Chig-gar-roo-gar-rem / Rough Tough! Real Stuff! Texas A&M!
On the other hand, for decades some boosters of the University of Texas have adopted the practice of shortening the wording of the orange decals that they put on the rear window of their car from "THE UNIVERSITY OF TEXAS" to "THE UNIVERSITY," as if to say that it is the only university in the state worthy of the name. And one clever person took this presumptuous attitude to an amusingly extreme degree by dropping the "ITY" and adding the "E" from "TEXAS" so that the decal read simply "THE UNIVERSE."