Does Jesus have a last name?

He had no last name. In the Middle East, it was (and still is) custom to name someone after from where they come. Like Saddam Hussein al-Tikriti. (Saddam was from Tikrit, Iraq). Hence the name "Jesus of Nazareth."

Sorry for mentioning the Savior of mankind and such an evil man in the same sentence, but I had to make a point.

Did Jesus have a last name?

Since you didn't specify if you referred to his human family last name, I will answer according to what name His Heavenly Father gave Him:

Philippians 2:9

9 Therefore, God elevated him to the place of highest honor and gave him the name above all other names, 10 that at the name of Jesus every knee should bow, in heaven and on earth and under the earth, 11 and every tongue declare that Jesus Christ is Lord, YHWH to the glory of God the Father

This scripture informs that because Jesus humbled Himself to become a man, and further to die the death of the cross for humanity, that the Father highly exalted Him and gave Him THE name that is ABOVE every other name, so that every human will bow and declare that Jesus Christ is LORD - which means YHWH Almighty God.

The Father gave Jesus the man, the LAST NAME of HIMSELF the GODHEAD - YHWH.

Jesus had the name before he came to earth, but he received it as the son of man/God, in this form, where he is now in heaven still in this form making intercession for us.

JESUS' LAST NAME IS YHWH, GIVEN TO HIM BY THE FATHER, WHILE JESUS WAS IN THE FORM OF THE SON OF MAN.

Up to this day, children of Semitic parents inherit the father's or grandfather's or from an ancestor a lot of generations back surname as their family name. In Semitic languages such as Aramaic, Hebrew or Arabic your last name refer to your father's surname with the prefix beth/bar, bet or bin, respectively. The prefix means 'house' or 'house of'. So Jesus is often described as being from the House of David. The lingua franca in Israel and in the Middle East was Aramaic at His time.

1 John 2:22 explains that the one who don't believe in Jesus that is the CHRIST, the translations from Hebrew to Greek is Messiah or anointing one that explain the last name Christ. The first name was given to Mary by the Angel.

Since in that day and age, your father and the place you come from, were used to describe someone...

He would have been called:

Yeshua ben Yosef haNotzri ~ Jesus son of Joseph of Nazareth

of course in everyday speech sometimes only:

Yeshua ben Yosef or Yeshua haNotzri

People did not refer to him as Christ (Messiah) in their every day speech, as most did not recognise Him as the Messiah, and it was dangerous to confess Him as such.

Jesus had no last name. Christ is a title that was given to Him. As noted in other questions, "Christ" comes from the Greek word "Christos", which is the translation of the Hebrew word "Meshiach", from which we derive the word "Messiah". So, "Messiah" and "Christ" are transliterated words for the same thing. These words mean "anointed one" in their original languages.

Last names or family names likely had their origin England about 800 years ago, but this was not common in Jewish cultures in the days of Jesus.

However, there were sometimes distinctions made for individuals based on their city of origin or residence. Jesus was sometimes referred to as "Jesus of Nazareth". He was not born there, of course, but that is the place Mary and Joseph lived after their return from Egypt. (Jesus was born in Bethlehem.)

Incidentally, this was the cause of some confusion about whether or not Jesus could be the Messiah, since the people knew that the Messiah would come from Bethlehem and not Nazareth, although another prophecy indicated that He would be called a Nazarene.

As David Morton noted, people were also distinguished by their parentage. This is seen in Simon Peter, in particular, whom Jesus referred to as 'Simon, son of John".

He brought him to Jesus. Jesus looked at him and said, “You are Simon the son of John. You shall be called Cephas” (which means Peter). John 1:42 ESV

In John 1:45, we find both of these distinctions in Philip's reference to Jesus:

Philip found Nathanael and said to him, “We have found him of whom Moses in the Law and also the prophets wrote, Jesus of Nazareth, the son of Joseph.” John 1:45 ESV

Finally, Jesus was also distinguished by his trade. He was known as a rabbi or teacher.

So, there was little need to distinguish between other people with similar names when referring to the "rabbi named Jesus", to "Jesus of Nazareth", or to "Jesus, the son of Joseph".