How do you change a name or address on an Ohio Concealed Handgun License?
You must notify the Sheriff’s Office that issued the license within 45 days after the address change. Check their website as some have a form that you can mail them for the address change. If you want a license with your new address issued the cost for a replacement is $15.[source]
Name changes may have to be made in person. We recommend visiting the Sheriff’s Office that issued the permit within 45 days of your name change to find out their process. The cost for the replacement license will be $15.
I found this info on this site in the section that deals with "Permit Information by State."