Did Claudius the Roman emperor have a surname?

His full name was Tiberias Claudius Nero. We just call him "Claudius" to avoid confusion with two other guys in his family.

To go with their awkward numbering system, the Romans had, in an age when most folks got by on just one name, a minimum of three: personal, clan and gens.

Clan and gens is the same (gens is the Latin for clan). The Romans used praenomen, nomen, and cognomen. The first one was the personal name. The second one was the gentile name (nomen gentile) that is, clan name. The third one was a nickname that distinguished people form other people with the same name.

For this emperor, we use the clan name because his personal name was Tiberius, the same name as the earlier emperor Tiberius (who was not a member of the Claudii, see below for this term) . Afterwards it goes back to naming by personal name as with Nero Claudius, which is shortened as Nero.

Claudius tells you a man comes form the gens Claudia, which can also be called the Claudii