score:18
It's hard to say for sure, but I strongly suspect it was named for the historic Palmyra in Syria. Most of the American settlements were only established (or renamed) after 1802 - skimming the WP articles, I get them as GA 1840, IL #1 1855, IL #2 unknown (but almost certainly later), IL #3 1814, IN 1839, ME 1807, MO 1819, NE 1870, NJ 1849, NY 1796, OH #1 1807, OH #2 1835, PA 1806-1810, TN unknown (but 1801 tax records mention Palmyra), UT 1852, VA 1814, WI 1846.
So only two of these are old enough to be the namesake of the ship Palmyra. But these were both far inland, so not likely to be closely involved with shipping, and neither had had that name for very long, so it was unlikely someone from there had moved to the coast, established a business, and named a ship after his hometown.
All the US towns named Palmyra that give a source for the name seem to have adopted it after the Syrian city; presumably it sounded nicely classical and faintly Biblical (even though the name itself isn't in the Bible, the city had associations with Solomon, etc). Given the wave of other things being named after it, it seems very likely this was the source of the name for the ship as well. But, of course, unprovable without a contemporary source.
Upvote:1
I'll second Andrew's answer that it was certainly ultimately from the ancient Syrian (≠Syria) town of Palmyra that served as Rome's entrepôt for Persian goods¹ and thus part of the trail end of the overland Silk Road connecting Antioch to Chang'an in Han China. (The actual border town was Dura-Europus but that name is much clunkier and never became a popular one for ships or US towns.)
Palmyra is that place where the route splits between Antioch and Damascus
It probably was directly named for the city as well. This ship was out of and owned by residents of Providence, Rhode Island,²³⁴ and intended—like many non-whaling New England ships of its era—for the Old China Trade. It was en route to Guangzhou ("Canton") at the time it passed the atoll,⁵ the First Opium War not having opened up any of the other treaty ports yet and China still imposing a form of the Ming haijin ("sea ban") after a brief period of greater openness at the onset of the Qing. I don't see any direct statements from the owners—e.g. from the Rhode Island Historical Society—but the city was very famous at the time as an exotic free agent connecting the commerce of empires, suiting the ship perfectly. William Pitt's niece went out her way to visit,⁶ it shows up in Shelley's Last Man,⁷ and Romantic readers ate up the story of its tragic queen Zenobia.⁸ Yes, it absolutely sucks that it's most well known now for having been destroyed.⁹ If it wasn't directly from the ancient city, it's more likely for one of the numerous other American, British, and Dutch ships that were.
Note that the US Navy at least mistakenly derived the island's name from palmyra, a species of palm whose name is owed to Portuguese palmeira by an entirely separate derivation.¹⁰