The right answer is indeed “son” because we say “un nom” and “un prénom” no matter if they are used to speak about a male or a female. Sa (fem.) son (masc.) and ses (plu.) are called in french “Adjectif possesifs”.

If in english we say :

“She walks her dog”

in french we will say :

“Elle promène son chien”

if the dog (chien) is a male or

“Elle promène sa chienne”

if the dog is a female.

It’s easy see the word after the possessive adjective or blank in your case and if it’s feminine singular then sa for masculine singular son and for plural it’s ses.