He does not have a first name or last name, he is simply known as Archimedes or Archimedes of Syracuse!.
In those days, the populations were not all that huge, (though they did have cities), and it was common for an influential person to be known as being of a particular city or region. In his case,...
chances of it being phidias.............. because his father was phidias................... not sure of this post.........................
The father's name was Phidias and he was an astronomer. His mother's name is unknown.
Archimedes was only known as simply that: Archimedes. He has no known surname of any kind. Otherwise, he's known as Archimedes of Syracuse.