The American Civil Liberties Union (ACLU) is currently suing the state of Ohio for their "refusal to correct the gender marker on birth certificates for transgender people, for any reason, at any time."
To change your name on your birth certificate you must,
Receive a legal name change order from a probate court.
Complete an application for a certified copy of one’s birth certificate with the Ohio Department of Health.
Attach the legal name change order to reflect the updated legal name.
The completed application and attachments must be mailed to: Ohio Department of Health, Vital Statistics, P.O. Box 15098, Columbus OH, 43215.
After the completed application and accompanying fees ($21.50) has been processed, the updated certified copy of the birth certificate will be mailed to you. Keep this on hand as you will need it for future filings.