2014 New York Laws
PBH - Public Health
Article 41 - VITAL STATISTICS
Title 3 - (4130 - 4138-D) REGISTRATION OF BIRTHS
4134 - Birth certificate; unnamed child.

 4134. Birth certificate; unnamed child. 1. If a living child has not
  yet been named at the date of filing the certificate of birth, the space
  provided  for  the  given  name  of the child is to be left blank, to be
  filled out subsequently by a supplemental report, as herein provided.
2. If a child dies without a given name, there shall be entered in the
  space provided for the name the words "died unnamed." The parents of the
  deceased child may submit a supplemental report of the given name of the
  deceased child. Such report shall be returned to the local registrar and
  the given name shall be entered on the original birth certificate.
3. When any certificate of  birth  of  a  living  child  is  presented
  without  the statement of the given name, the local registrar shall make
  out and deliver to the parents of the child a supplemental report of the
  given name of the child, which shall be  filled  out  as  directed,  and
  returned  to  the  local  registrar as soon as the child shall have been
  named. The given name, supplied by the  supplemental  report,  shall  be
  entered on the original birth certificate.