Yes, they did, with the Argo, traditionally thought in most versions of the story to have been the first sailing ship, being the most famous. More historical names can be found in the Tabulae Curatorum Navalium, a series of inscriptions, mostly from the 4th Century, at Athens which catalogs all the naval equipment and stuff in the city's fleet, including a list of the names of several hundred ships.

Ships have been named as far back as we can tell to the third millennium BCE. They were typically named after gods typically to try to garner their favor for dangerous trips. The Argo from "Jason and the Argonauts" was names after its builder Argus and also means "swift" in Ancient Greek. Ships are also referee to as "she" in most languages (The Argo rocked about as she hit the wave.) regardelss of the gender of the name for a number of possible reasons. Sailors were normally men and reminded them of women at home, the latin word for ship was feminine, and a few others.