This is how Bart Ehrman tries to trick his students: by asking the question, "What was Jesus' (or Paul's) last name?" The fact is that people in Antiquity did not have last names. They were usually identified as "[name] son of [the person's father's name]" or "[name] of [place of birth]." This is why Jesus is often referred to as Jesus of Nazareth. (There is strong evidence indicating he wasn't born in Bethlehem.) At times, he is also called Jesus son of Joseph (John 6.42) and Jesus son of Mary (Mark 6.3).
In addition, in Antiquity, some people had sort of nicknames, or cognomina (cognomen in singular form), given to them, which is why Jesus is most often called Jesus Christ (the latter part of his name being the cognomen.) A cognomen was initially a nickname given to someone because of a certain characteristic this person exhibited. This characteristic was supposed to differentiate the person from others. Since Jesus was known among his disciples as the anointed one (christos in Greek, maschiah in Hebrew, Christ/messiah as English transliterations), he was given the cognomen christos. It was later translated into Latin christus, which then was transliterated into English Christ.
Having said that, I am convinced Jesus' parents did not give him this nickname. In our earliest gospel, gospel of Mark, Jesus' family thinks that he is out of his mind because he began to preach (Mark 3.21)! (This verse also indicates that the author either didn't know of or didn't believe in the virgin birth story. This has led scholars to believe that the virgin birth story was a later invention.)