Playing up "New Jersey" isn't a new idea, although they haven't been consistent about it over the years.
Rutgers started using "New Jersey's Team" as a slogan in 1984. At that time, they put NJ above Rutgers on the helmet. One end zone said "State University," the other said "New Jersey." The center of the basketball court got ridiculously wordy, with "The State University" and "of New Jersey" surrounding the word Rutgers.
Sports articles would use the occasional reference to "the state university," or sometimes "New Jersey's state university." New Jersey pride was riding very high in the 80s.
Bill Cosby and Gov. Kean each did commercials touting the state (though these days, The Cos isn't a great endorsement).
New Jersey and You - Perfect Together
Closer to the present, during Greg Schiano's first run as coach, there was a TV commercial with him and James Gandolfini of "The Sopranos," with a fan coming up to them talking about his great work and "the Jersey thing going on," the semi-obvious twist that he was talking not about the Sopranos, but Rutgers football.
Schiano took it the other way, making his recruiting grounds "The State of Rutgers," which was New Jersey, the nearby states, plus others like Florida in particular.
Rutgers IS New Jersey's college team, so I'm all for promoting statewide interest. But I think there's way too much invested in the Rutgers name to change it.