"She will bear a son, and you shall call his name Jesus..."
Matthew 1:21a (ESV)
Mary and Joseph did not get to pick the name of their first born son. Had this been a traditional pregnancy following the betrothal of Mary and Joseph, Joseph more than likely would have called Jesus, "Joseph" after himself. Zachariah's family members wanted Elizabeth to name her baby, whom we know as John the Baptist, after his father. Interestingly enough it seems Mary and Joseph did not have last names. This might make it hard to look them up in the Nazareth phone book.
The first people to have used a "surname" or last name was the Chinese. The story goes that an Emperor named Fushi decreed the use of surnames or family names around 2852 BC. The Chinese people typically have three names. The surname, or last name comes first, that name is followed by a generation name, and then placed last is the given name, or the first name.
The Romans for most of their history had only one name. However, they eventually began using three names. The given name came first, as a first name, this was followed by the name of their clan, and the last name listed was to designate the family. History shows as the Roman empire declined the use of multiple names began to be discontinued and eventually there was a return to single names.