http://www.sos.state.oh.us/sos/upload/elections/forms/11-A.pdf makes it look like one can get an absentee ballot in Ohio with the last four digits of one’s social security number.
Otherwise, why can’t he get a bank account with statement address at his college location, or an Ohio driver’s license? Fully online bank accounts produce online bank statements that can be printed out. (Some such bank accounts have no minimum balance and no fees.)
Yes, Ohio seems to be deliberately trying to make it difficult for college students to vote. The best way to protest its policy is to vote there.