The Ancient Greeks certainly did!
For warships or transports there doesn't seem to have been a uniform naming convention necessarily, though the gods and their children as well as the location of where the ship was built seem to be among the most popular themes. On the other hand, there were several ships we know of that were so-called sacred ships that had important functions in addition to being necessary for the naval fleet. As far as the evidence we have is concerned these sacred ships seemed to have been an Athenian practice.
Among the names of ancient Greek ships we know of:
As far as the Romans are concerned, they very commonly named their ships with casual convention. The gods, great heroes, statesmen, places, battles, historical events, themes, ideals, ideas...there were a full range of names that the Romans assigned to their ships. I don't have full access to my texts at the moment since I'm at work but I'll happily add in some names as examples here later.
Yes. The Egyptians already named their ships, as did the Athenians and other Greeks (as /u/ANovelEromenos has already pointed out above), and, same as them, the Romans named their ships. These names came in great variety, and no attempt seems to have been made to name certain classes or sizes of ships after a certain type of name, so there was much variation among the fleets and ship-types. Roman soldiers often proudly displayed the units and sub-units, such as centuries or wings of cavalry, they served in on their tombstones, and similarly, Roman sailors would do the same for the fleets and ships. This is one of the ways how we know a large number of Roman military ship names. Here is an example from Saturnia, Etruria (AE 1978, 311):
L(ucius) Lucretius / Firmus mil(es) / ex classe praetoria / Misen{i}ensis ex lib(urna) / Iustitia 7(centuria) C(ai) Minuci / Auxiliari(s) vixit an(nos) [------
'Lucius Lucretius Firmus, soldier (sailor) from the praetorian fleet of Misenum, of the Liburna (a smaller, nimble type of ship) 'Iustitia', in the century of Caius Minucius Auxiliarus, lived (?) years....'
Another example would be this inscription, from the tombstone of an egyptian Sailor, also serving in the fleet of Misenum, from Sarmandag/Seleucia Pieriae in Syria (AE 1939, 222):
D(is) M(anibus) / Iulius Deme/trius mil(es) cl(assis) / pr(aetoriae) Misen(ensis) III(triere) Pietate nat(ione) Aegypt(ius) vix(it) an(nos) / XLV mil(itavit) an(nos) XXV / h(eres) b(ene) m(erenti) f(ecit)
'To the infernal spirits [of] Iulius Demetrius, soldier (sailor) from the praetorian fleet of Misenum, of the Triere (ship with three banks of oars) 'Pietas', an Egyptian, [who] lived 45 years and served 25 years, the heirs made this [for the] well-deserved.'
Like I said, there was a wide variety of names, and no real system to the distribution of names among the naval units. There were several categories of names that seem to have been in wide use:
One exception are the names of rulers, of which apparently only Augustus was used. Another peculiarity which is different from modern or even Greek practice is that ships are not feminine. Masculine forms of names are used as often as feminine ones.
These names were represented as a scultped or painted device, the episemon or parasemon, on the sides of the ship's prow, in the likeness of the entity the ship was named after, which would make the name easy to discern from the distance, even without any kind of telescopic aid.
This practice was common enough that a ship could be described, or named, after the parasemon that it carried, such as in this votive inscription, erected by a slave who was in charge of accounting for a ship's captain:
... Dionysio Sostra/ti filio Alexandrino gubernatore / navis parasemo Isopharia T(iti) Cl(audi) Theonis
'[accountant for] Dionysios Alexandrinus son of Sostratos, captain of the ship of the name-device Isopharia (Isis of Pharos), [of] Titus Claudius Theonis'
St. Paul was carried in a ship cui erat insigne Castorum (Acts 28, 11), a ship from Alexandria which bore the device of the Castores (i.e. Castor & Pollux).
Merchantmen, of which we have less examples, seem to have been named mostly after deities, particularly popular on the Alexandrinian route was, as above, Isopharia, the Isis of Pharos (the isle of the Alexandria lighthouse), which seems pretty fitting - a prominent starting point and a sight that must have been most welcome after an often perilous journey across the mediterranean. In a similar vein, the name of Fortuna Redux might have been popular as well, the tutelary deity of safe return. According to a papyrus from Panopolis dating from the 4th century, it seems to have been customary to name ships after the tutelary deity of the respective city:
καὶ κ̣ὰν [γε β]ο̣υλόμενον ᾖ τοῖς ναυκλήροις ἐν̣ ταῖς πρῷραις τῶ[ν πλοίων] ὡς ἔθος ἐστιν τῶν ἐν ἑκά̣σ̣τῃ πόλει θεῶν τὰ παράσημα ἐνχαραχθή̣τ̣ω τοῖς πλοίοις πρὶν κατασπασθῆναι αὐτά
"if desired, shipowners, before launching their craft, may carve on the prows of the ships, as is customary, the devices of the gods of each city" [P.Panop. Beatty 2, 208 f., transl. Casson, 344].
It doesn't seem that too much variety among ship names was something that was sought after, or at least ships with the same name posed no problems, as seen above, they were usually also identified by providing the name of their owner. Ships without a name were called asemon, so having a name was not a necessity, though it's hard to say how many ships didn't have a name. It seems that the names chosen also may have had apotropaic function, so sailors probably would have wanted to have a name that brought luck. The gravestone of the wife of a frumentarius of the 10th Legion 'Gemina' from Carnuntum, Pannonia Superior in modern Austria, bears the relief of a small ship apparently named 'Felix Italia', lucky Italia, here a reproduction from the local museum, Casson holds it more likely that it was more to identify the ship in the relief thatn representative of actual words carved into the boat itself.
Sources:
L. Casson, Ships and Seamanship in the Ancient World (Baltimore 1995), mainly p. 355-360.
Epigraphic/Papyrologic sources given directly in the text.
Absolutely. For example we have the Tabulae Curatorum Navalium in Athens, a 4th BCE century naval list of all the ships in the Athenian Navy of the time. And most of the names (translated) would be familiar in most navies today. Here are some of the common naming conventions.
The Romans also named their ships. For example in the early Roman Principate Augustus established a fleet in the bay of Naples to control the Western Mediterranean. These showed that the popular naming themes remained popular: Gods & People: Venus, Hercules, Augustus. Places: Tigris, Euphrates, Danubius, Tibur. Concepts: Fides, Fortuna, Libertas. The Romans though didn't maintain the obsession with keeping with feminine forms for ships. They were perfectly fine with masculine names.
I mean, there's always the Argo. Casson notes that the Argo is the only ship whose name we know until the 4th Century, by which point we have the Athenian naval lists preserved. That's not totally right (Thucydides mentions a couple of ships by name, notably the Σαλαμινία), but the Athenian naval lists provide the names for around 300 warships, some of which were repeated by several ships. There's no particular system to the naming of ships, except that Greek ships appear to have been normally feminine. So we get ships whose names are adjectival (Χρύση, "golden," Λαμπρά, "bright"), we get ships whose names describe physical qualities of the ship or of women (Παρθένος, "virgin," Nέα, "new, young"), we get ships whose names are indicative of speed (Πετομένη, "flying," or Argo herself, which means "swift"), we get ships whose names are warlike or frightening (Ἀνδρεῖα, "courage, manliness, virtus," Δύναμις, "power"), we get protective names (Συμμαχία, "alliance," Βοήθεια, "help, assistance"), we get animal names (Ἁλκυών, "kingfisher," Λέαινα, "lioness"), we get ships named for weapons or martial equipment (Αἰχμή, "spear"--in Homer this word actually means "spearpoint" if I remember right--Σάλπιγξ, "trumpet'), we get place names (Σαλαμινία and Παράλια, the two messenger-ships, as well as a number of ships named after demes or regions of Attica, or even Athenian subjects and allies), we get names invoking luck (Εὐδαιμονία, "good fortune," Εὔπλοια, "good sailing"), we get...you get the idea. A lot of names on the naval lists are abstract nouns, such as Δημοκρατία (you can guess what that means), Ἐλευθερία (freedom), Σοφία (wisdom), or even Τραγῳδία and Kωμῳδία. Casson says that a particular favorite of the Athenian naval lists was Νίκη (victory), and names formed by compounds of that word. Besides Nike there are a relatively small number of divine names, mostly names like Θέτις or more usually adjectival names either composed of an epithet to a goddess or a feminized form of a hero's or god's name (Ἀχίλλεια, "the Achillean," or Κυθέρεια, "Cytheria, one of the cult titles of Aphrodite). You'll notice that some of these names are not nouns or adjectives at all, but are verbal: Πετομένη, or Νικῶσα (winning) are both present participles. The use of participles to name ships is not especially uncommon, and there are (exactly two, as I recall) ships whose names are finite verbal forms. One is a form of Nike (Νικήσω, "I will win," the 1st person sg fut ind of νικάω) and the other is Ἡγήσω, the aorist middle imperative of ἡγέομαι, "to lead the way" (this name, kind of bizarrely, was apparently an accepted girls' name--a famous grave stele from the late 5th Century bears this name).
Casson remarks that our evidence for Hellenistic ship names is pretty spotty, but they appear to have generally followed the same pattern, although several of the names that Casson mentions (Ἡρακλῆς, Ἀσκληπιός) are masculine forms, from the names of heroes or gods. Roman ship names are very well-attested from epigraphy, and follow many of the same general patterns, although the Romans have more of a tendency than the Athenians to use masculine names and name ships after deities. And in the case of Roman ships these aren't just the names of warships, as they are for the Athenians. Roman sailors and ship owners often included the names of their ships on their graves, so we know of ships with names like Fortuna Redux ("Fortune returns," i.e. Fortune that brings one back into port safely) that would be pretty inappropriate for a warship but perfectly fine for a cargo ship.
