Yes, the ship of Theseus was named, or at least that is what we are told later:

The Athenians had a special type of ship called a theoris (θεωρίς), which was used for religious missions and state business. There were three that we know of. Two of them, called the Salaminia and the Paralus, were named after the origins of the crewmen who had originally sailed them. We hear of them most often in the 5th century, when the Athenians sent them here and there as special warships, as messenger ships, and so forth. Famously, the Salaminia came to fetch the general Alkibiades home to stand trial in 415 BCE. (Thucydides VI.53)

The third was called the Delia, and the Athenians in historical times believed it was the original ship which had carried Theseus back to Athens over the waves from Crete. Whether you believe that or not, the ship was certainly old, and often hear that it was constantly repaired to keep it seaworthy. Its main (and seemingly only) mission was to make a religious voyage (the θεωρία) from Athens to Delos during the festival of the Lesser Delia, at which time the priest of Apollo adorned it with laurel garlands. While the ship was gone on this mission, Athens was in a state of ritual cleansing. Supposedly, the Lesser Delia was established to celebrate Theseus' slaying of the Minotaur, but it was obviously centered on a ritual procession related to Apollo, and after all, the ship went to Delos (birthplace of Apollo) instead of Crete. The ship Delia was used as an example of something which was both old and new at the same time (because by the 5th century, it was so old that all its planks had been replaced). Plutarch mentions it briefly in his Life of Theseus, but there is no further information, except that it apparently survived until the time of Demetrius of Phaleron, in the last quarter of the 4th century BCE. (Plutarch Thes 23).