As I former Ohioan, I was always told that they went by "The" to differentiate themselves from Ohio University and to let everyone in the state know that they were "the" premier university in Ohio. During the 90's and later there were many lawsuits between The Ohio State University and Ohio University over the use of the word Ohio and I believe one was centered around The Ohio State University trying to trademark the use of Ohio and especially the famous script Ohio. There is a lot of contention between the two schools and their supporters.