In Ohio, the process of changing or correcting a birth certificate that has already been issued is handled by the Probate Court in your county.
The process is covered under Ohio Revised Statute 3705.15 which reads:
"Whoever claims to have been born in this state, and whose registration of birth is not recorded, or has been lost or destroyed, or has not been properly and accurately recorded, may file an
application for registration of birth or correction of the birth record in the probate court of the county of the person's birth or residence or the county in which the person's mother resided at the time of the person's birth. If the person is a minor the application shall be signed by either parent or the person's guardian.
(A) An application to correct a birth record shall set forth all of the available facts required on a birth record and the reasons for making the application, and shall be verified by the applicant. Upon the filing of the application the court may fix a date for a hearing , which shall not be less than seven days after the filing date. The court may require one publication of notice of the hearing in a newspaper of general circulation in the county at least seven days prior to the date of the hearing. The application shall be supported by the affidavit of the physician in attendance. If an affidavit is not available the application shall be supported by the affidavits of at least two persons having knowledge of the facts stated in the application, by documentary evidence, or by other evidence the court deems sufficient.
The probate judge, if satisfied that the facts are as stated, shall make an order correcting the birth record, except that in the case of an application to correct the date of birth, the judge shall make the order only if any date shown as the date the attending physician signed the birth record or the date the local registrar filed the record is consistent with the corrected date of birth. If supported by sufficient evidence, the judge may include in an order correcting the date of birth an order correcting the date the attending physician signed the birth record or the date the local registrar filed the record."
Source:
http://onlinedocs.andersonpublishing.com/revisedcode/jumptext.cfm?jumppage=3705.15
You must also pay the allowed court costs for this filing. Contact your county probate court for costs and if you have any questions.
You might take a look at the Franklin County Probate site for some general information:
http://www.co.franklin.oh.us/probate/departments/Certification.htm#Birth correction