But nobody really knows where “Oregon” comes from.
No. For Louise Caroline Alberta, fourth daughter of Queen Victoria.
Lake Louise in the Albertan Rockies is also named after her.
Three provinces of Panama qualify:
Colón, after Columbus
Herrera, after Gen. Tomás Herrera
San Blas, after the Saint
Even if it was intended to honor James, as “New York” surely it was named after York itself, rather than its Duke?
No. Titles and the places they refer to are quite separate - it’s no more named after the city than my earlier examples of the Falklands and South Sandwich islands are named after those towns. (FWIW, the current Duke of York is Prince Andrew, who has no particular connection to the place)
Catherine of Oregon? 
From what I remembered from my history lessons, China has always been “Zhong Guo”, due to its isolation from the rest of the world by the deserts and mountains, giving the impression that it is at the centre of the known world.
Granted, “Zhong Guo” then doesn’t to the entire region claimed by country now – usually it means the ‘middle plains’. Here’s a cite from Wikipedia: China - Wikipedia
I’d thought Turkey was named after Ataturk. See from Wikipedia it’s got a different origin. Guess that’s the one new thing I learnt today.
Russia was named for Rus, the Viking founder of the nation and of the Rurik dynasty.
Can we have a cite for that? I blieve the people were called “Rus” as a whole-- it didn’t refer to a particular individual. I don’t think there is even agreement that word “Rus” is Indo-European in origin since it might derive from Finnish.
An interesting bit of history in the formation of the nation of Liberia in Africa has a town named Clay-Ashland in an area known as “Kentucky in Africa.” Established in 1846, it was named in honor of statesman Henry Clay and his Lexington, Ky., estate, Ashland.
…twinned with Legoland
This fails to actually answer the question. Because the name is New York, it implies that it is named after a place, not a person. Otherwise you would expect the name of the city to be Yorktown, or just York City, and the state to be Yorkland or Yorkiana. If the name refers to a person, then it implies it is named after some new Duke of York, not James.
The construction is similar to that of New Hampshire and New Jersey, which are named after places. New Jersey is named after the island of Jersey in the Channel Islands because of the assistance it gave the Stuarts during the English Civil War. It differs from other states named after people, such as the Carolinas, Pennsylvania, Maryland, or Louisiana.
Can you explain the background of the construction “New York” if the state is specifically named after the Duke of York?
Because it was a renaming of the conquered New Amsterdam.
…oh, just discovered that Albany was so renamed at the same time, after the Duke of Albany.
British royal titles (and some noble ones as well) are derived from historic toponyms (ancient kingdoms and principalities, shires, cities, etc.) but have no direct connection to the site used. The Dukes of Kent and Glouester have no particular conection to the shires that gave rise to their titles. Information Please Almanac’s website (not my favorite choice of cite, but convenient here) confirms Gorilla Man’s and my statement about it being the Duke of York, not the former city of Eboracum, for which New York was named. However, to avoid confusion between the newly-acquired colonial city and the ancient English city which was the source of King James’s title while King Charles II’s brother and heir, one presumes the “New” was added for clarifying purposes.
One wonders if either James II or Prince Andrew had 10,000 men to march up and down hills. (In passing, what Duke of York was it who did? Edward IV’s father? And has Randy Andy had 10,000 women yet?)
I got it from a book I read in high school 30-odd years ago, so unfortunately the title and the author’s name are unavailable. The online sources I looked for call him “Rurik, founder of the Kievan Rus,” reaffirm that he was a Viking and posit that he may have been more legendary than real.
Nope - he was George III’s second son, Frederick, Duke of York and Albany. The nursery rhyme refers to an episode when he was in charge of a British army in northern France or Flanders during the early Napoleanic wars.
for a somewhat frivolous further discussion on this point, see post 23 of this thread. 
While we cannot be completely sure, at least the Finnish agree, that it was the people of Scandinavian origin who were called Rus, who gave their name to Russia. It’s either possible that Rurik was a real person or that he’s just a legendary figure who resembles the actual leader(s) of the Rus, who were Varjags/Vikings themselves. But Russia is named for the Rus and not for Rurik, whose name is etymologically same as Roderick and is not related to the name Rus, but simply means something like “famous ruler”.
The most likely reason why these Vikings (Nestor’s Chronicle uses the word Varjag or Varangian) were called Rus is that they are named for Swedish region of Roslagen. During 800s Roslagen was the name of eastern coast of Svealand (Central Sweden) and the archipelago. Its name comes from something like “Ros”+“lagen” where Ros is short for Roden, another name for Hundreds which were used as geographic division within Germanic peoples, and lagen means law. As Roslagen was the eastern coast, the Swedish Vikings traveling east started their voyages from there. When they contacted the Finnic peoples living in what are now Finland and Estonia they must have called themselves ‘Ros’ or ‘Rus’ as even today Sweden is “Ruotsi” in Finnish and “Rootsi” in Estonian. Once this was established, it was natural for the Finnish to call all their western visitors “Rus” even if they weren’t from Roslagen.
After 850 or so, some of these Scandinavians have established themselves in what is now Northern Russia so that Rurik rises to be a leader in Novgorod. There were some Slavic tribes already living so far north, but most of the area was still Finno-Ugric. At this point the name Rus seems to become the name of a nation, as those Finno-Ugric tribes south of Lake Ladoga use it and the Baltic Finns notice that there are the same Rus now living in both west and east of them. Rurik’s and his follower Oleg’s Rus then take Kiev and form Kievan Rus, which will later lead to Russia. Thus the name Russia originates from Swedish but it’s possible it was popularized by Finnish.