If the goal here is to correct mom's last name on the birth certificate, as a disparity of the type you describe would not be obvious, "A change of name ordered by a court of competent jurisdiction shall be required to change the name as shown on the birth certificate, unless the registrant, parents, legal guardian, or individual responsible for filing the birth certificate presents documentation that the name was incorrectly recorded at the time of registration of the birth and meets the requirements in Section 8 of this administrative regulation." Mom can contact the Department of Vital Records to inquire about the forms and process for correcting her name without going through court, if possible under the full facts.
If the goal is simply to change the baby's name, that process is summarized here.

