The desert tribes of the Middle East in the first to second millennium BCE didn't have last names. The Judeo-Christian god does have a name, which many of them seem to forget. Jehovah was a minor sky deity of the Canaanites, one of the seventy sons of their Supreme deity El Elyon. When the Hebrews broke off they took Jehovah as patron.
I don't know when last names came into use in the Hebrew Tradition---the use of last names overall has developed at different times in different cultures, so there's no one answer for that I think.
In regard to the traditions still in use in Jesus' time, the "last name" so to speak was "ben <insert father's name>" I've also seen other name structures from the time that use "bar" instead of "ben". Both of these mean "son of ...?"
Thus, Jesus would have been known as Jesus ben Joseph. For Joseph, it's unclear--the genealogy in Matthew says his father was called Jacob; but Luke writes that Joseph was a son of Heli, so it's either Joseph ben Jacob or Joseph ben Heli.
While a woman's "last name" would then supposedly followed suit ("bat" for daughter of), unlike for the men, I personally have not seen any translations of the Bible or contemporary works that used that formula. I've only seen designation by association with a hometown or geographic area (for example, Mary Magadalen was called that because she was from Magdala, The Virgin has been referred to as Mary of Nazareth, etc)
God being a title, you first need a first name. Now the biblical title for God is El, so the God of the bible used a few names, (in fact robbed his fathers name Elyon in a few verses) Shaddai, Elyon, and Yahweh (also known as Jehovah) the last one is the most commonly referred to in the bible. But as Skeptikitten points out ancients did not use last names and being this God was dreamed up by ancient herders they would have never thought up a last name.