Jesus full name in Hebrew is 'Yeshua bar Yosef' which translates to Jesus son of Joseph. The name "Jesus" is derived from the Greek "Iesous", which in Hebrew translates to "Yehoshua" which is rendered "Joshua."
I like daves answer but will add this cut & paste from an article.
The Judaic practice of the time was to call a man after his father's name. So: •According to Matthew's Gospel. Joseph would have been "Joseph ben ('son of') Jacob" (depending on the Aramaic form for Joseph). •According to Luke's Gospel, Joseph would have been called "Joseph ben Heli".
People didn't really have last names then. Joseph was the son of Jacob. So if Joseph had a "last name" his name would have been kinda like "Joseph Jacobson." In the same way, Jesus would have been known as Jesus Josephson. This mostly applies to when people were in their home towns. Then everyone knew who "Joseph" was. When people traveled they were normally identified by their home town. That is why Jesus is often called "Jesus of Nazareth." The "Of Nazareth" is also essentially his last name. It distinguished him from all of the other Jesus' of the era, and by all accounts it was a popular name
Last names throughout most of history were the father's name. Jesus was Jesus bar Joseph or Jesus ben Joseph. The actual Hebrew and Aramaic didn't actually have J's though, so Jesus would have been Yeshua bar Yosef or Yeshua ben Yosef.