The official title of the college is University at Buffalo, The State University of New York. SUNY-Buffalo is one such abbreviation, or "University at Buffalo," but around these parts we just call them "Buffalo." The Buffalo Bulls is a name perhaps confused with "Buffalo Bills" if you're a fast reader, but it's a well-established identity in college athletic circles. However, the name New York has way more traction in sports, media and revenue than Buffalo.
This Buffalo News blog item teases the hell out of a rebranding effort within the athletic department, code named "New York Bulls". According to Bob DiCesare, one of White's visions is to emphasize "New York" in the athletic department, as opposed to Buffalo. DiCesare tells an anecdote about how a businessman approached the school, offering to pay for a name change to New York State University. I don't know if this is a similar venture, or even the same one.
But I do know that there are basically two New Yorks; the city and the rest of the state. The name, without the context, usually brings my mind directly to the city, but if the team's name changed to something like, say, New York-Buffalo or NY-Buffalo, then probably not so much. We would still be calling it "Buffalo," unless their style guide strongly recommended us not to.
There are 22 Division I schools in New York, but just three are in FBS (Buffalo, Syracuse, Army). Moreover, there are four schools in the SUNY system in Division I (Buffalo, Binghamton, Albany, Stony Brook) which makes UB the only FBS one. It's also the largest school and the biggest athletic department. If White tried to make Buffalo the signature state school of New York, it could possibly help the visibility of the program. But it wouldn't necessarily make it the de facto major state school, because Syracuse ain't seceding any time soon.
Having said that, here are a few suggestions for UB's rebranding:
• The Big Apple Bulls
• Empire State Bulls
• Buffalo (of New York (of Ohio))
• New York Yankees
• Western New York Fighting Buffaloes
• New York Buffalo Buffalo Buffalo Buffalo Buffalo Buffalo Buffalo
• New York Yankees