Ohio has no registration requirements for handguns like Michigan. State level, for Ohio, there's nothing to do. I do not know if there is a legal requirement for a person who bought a handgun in MI to be able to show they have divested themselves of the weapon.
It is legal to give a firearm as a gift with no further action taken in Ohio. Again, not sure about MI.
It is also legal Federally to give a firearm as a gift in person across state lines. I'm not sure about getting it in one state, and taking it back across. Sort of sticky. If you really want to cover your bases, it should be transferred to an FFL in Ohio, and then transferred to her. That would be the best way to make sure you are covered.