Presumptively courts see it as the last name the custodial parent gave the child, is in the best interest of the child. However, that presumption is rebuttable through a preponderance of evidence showing why it is in the best interest of the child to have the father's last name.
Usually a suit for change of name/minor child, can be successful when the petitioner is the father. Courts tend to view a child having the fathers last name, who plays or wants to play an active role in the child's life, as in the best interest of the child.
Preponderance is simply putting forth evidence just enough to outweigh the case against what you're seeking.
As far as who takes what name in a marriage, that's to be decided by the marrying parties.