New Brunswick, named in 1784 - named for King George III who also held the title Duke of Brunswick-Lüneburg at that time.

New Jersey, named in June 1664 - named by James, the Duke of York, when he granted the land to Lord John Berkeley and Sir George Carteret. The name was a nod to Carteret who had previously served as Governor of Jersey off the coast of England.

New York, named in March 1664 - named by King Charles II as a nod to his brother, James, the Duke of York, to whom he granted the colony.

Previously parts of these two colonies were known as New Sweden and even earlier New Netherlands. That name first appears (as close as I can tell) in Oct of 1614 with the creation of a joint-venture in The Netherlands to oversee the colony. It may have been as early as 1609 when Henry Hudson claimed the land, but nothing was established permanently within it until 1614. In 1624 the colony of New Amsterdam (now NYC) was established.

New London, named in 1658 - Started in 1646 it was originally Pequot, named for the local people by the same name. Colonists pushed to change the name and in 1658 it happened.

New Hampshire, named in 1629 - named by John Mason after splitting the colony he (and others) had began from a 1623 grant (that colomy was eventually called Dover). Named for Hampshire, his home county in England. Also, Portsmouth, NH was named for the English town in which he served as fort commander.

New England, named between 1614 and 1616 - Captain John Smith wrote a [book] (https://babel.hathitrust.org/cgi/pt?id=nyp.33433113859981&view=1up&seq=9) called A Description of New-England in which the first use of New England appears, both on map and in text. On page 33 (by the link) you can see a number of locations he has renamed, with old and new names, though he doesn't directly add "new" in any of their names. He does later refer to New Plimouth, which he describes as "excellent good harbor, good land; and now want of any industrious people." It was simply coincidence that the port of call from which the Pilgrams sailed was also named Plymouth as it bore the name previous to their landing there. I say between 1614 and 1616 because he arrived in New England to explore and chart those maps in April of 1614 but they were not published until 1616.

After all that, we're still about 100 years off from the origination in the new world. A map had appeared many decades earlier, in 1529, bearing the name "Gallia Nova" or (basically) New France. They can't take the title to being first, however; in August of 1521 the Spanish conquistador Cortes took Tenochtitlan, subjugating the Aztec empire by a new name, Nueva Espana - New Spain.