According to the California Health and Safety Code (HSC), Division 102. Vital Records and Health Statistics, Part 1. Vital Records, Chapter 3. Live Birth Registration, Section 102405:
For live births that occur in a hospital, or a state-licensed alternative birth center, ... the administrator of the hospital or center or a representative designated by the administrator in writing may sign the birth certificate certifying the fact of birth instead of the attending physician and surgeon, certified nurse midwife, or principal attendant if the physician and surgeon, certified nurse midwife, or principal attendant is not available to sign the certificate; and shall be responsible for registering the certificate with the local registrar within the time specified in Section 102400. (10 days)
Section 102415 provides:
For live births that occur outside of a hospital or outside of a state-licensed alternative birth center, ... the physician in attendance at the birth or, in the absence of a physician, the professionally licensed midwife in attendance at the birth or, in the absence of a physician or midwife, either one of the parents shall be responsible for entering the information on the certificate, securing the required signatures, and for registering the certificate with the local registrar.
The code does not say whether a physician, midwife, hospital administrator, or other person who fills out a birth registration should get name information from mother, father or any other specified person. It seems not to contemplate a dispute on the matter.
HSC code section 102425
(a) The certificate of live birth for any live birth occurring on or after January 1, 2016, shall contain those items necessary to establish the fact of the birth and shall contain only the following information:
(1) Full name and sex of the child.
(2) Date of birth, including month, day, hour, and year.
(3) Place of birth.
(4) Full name, birthplace, and date of birth of each parent, including month, day, and year, and the parental relationship of the parent to the child.
(C) If the parents are not married to each other, the father’s name shall not be listed on the birth certificate unless the father and the mother sign a voluntary declaration of paternity at the hospital before the birth certificate is submitted for registration. The birth certificate may be amended to add the father’s name at a later date only if paternity for the child has been established by a judgment of a court of competent jurisdiction or by the filing of a voluntary declaration of paternity.
(6) Signature, and relationship to the child, of a parent or other informant, and date signed.
[The other contents of the certificate are omitted as not relevant here.]
HSC code section 103225 says:
Whenever the facts are not correctly stated in any certificate of birth, death, fetal death, or marriage already registered, the person asserting that the error exists may make an affidavit under oath stating the changes necessary to make the record correct, that shall be supported by the affidavit of one other credible person having knowledge of the facts, and file it with the state or local registrar.
HSC code section 103230 says:
Section 103225 shall be applicable to certificates of birth only in the absence of conflicting information relative to parentage on the originally registered certificate of birth.
It would seem that sometime in the labor process the mother could inform the physician and hospital staff what name she wants on the birth certificate. But the suggestions in the comments about consulting a lawyer in advance, one who is employed by the mother alone, seem very well taken in the situation described in the question.
The amendment process of Section 103225 could be used after the fact, but it does not seem well designed for contested circumstances. Indeed it seems mainly intended to correct clerical errors, and to change earlier certificates that insisted on a "father" and "mother" for a same-sex couple, or for a single father employing a surrogate. It is not clear what would happen if a mother filed a notice of correction under Section 103225 and the father opposed it.
Who provides financial or emotional support for the child, or who undergoes the pains of birth, are not apparently relevant under CA law to who selects the name of the child.