You're going to have to be more specific; what state, what school district.
If you live in one of those Ohio school districts that is primarily in the adjacent county, so it's not showing up on the drop down list, just pick any other district (preferably one without an income tax). It doesn't matter what you put, it's just used, by the state, for statistics. It's a flaw in Turbotax and they have no plans to fix it.
If during the review, there is a diagnostic asking you to enter in the school district but you can't enter it on that page, just go back to the inputs. Go back to the state in question and go through the residency section of that state. It should pop up again.