It looks like I might do one of these each morning for a few days, while I’m off work.
I gave myself five minutes and listed every American place with I could think of with “New” in the name. I got eight. See how many you can get. My answers are after the jump.
New York (England)
New Jersey (England)
New Hampshire (England)
“New England”
New Mexico (Mexico. Duh.)
New Haven (CT) (England)
New Bedford (MA) (England)
New Orleans (France)
NOTE: I also wrote “New Amsterdam,” but since that’s an outdated, historical name, I didn’t count it in the final total. New England, however, is still a specific place name in current use, so it counts. I included what nation the original location is in–note that most American “New” cities are English–or at least the ones I thought of. I wanted to include New Brunswick and Nova Scotia (New Scotland), but those are Canadian.
Gently Hew Stone 
Arkansas
New Hope township, Izard County, Ark (just a new start, I guess)
New Tennessee, Perry County, Ark (Ousted from Tennessee, too?)
California
New Helvetia Sacramento and Yuba Counties, Cal (Switzerland)
New Hope Track, San Joaquin County, Cal
Connecticut
New Canaan, Con (Biblical theme)
New Fairfield, CT (England)
New Britain, CT (Duh)
New Milford (England)
New London (there are Londons all over the US and the English-speaking world, either “New” or something)
Delaware
New Castle County and township (I wonder if it’s just a random castle or just variant spelling of Newcastle?)
DC
New Troy (Greece)
New Scotland Hundred (Administrative Divisions)
Illinois
New Trier (Germany)
New Haven (I’ve always thought this is, too, a place to start over, a “haven” in a world that is hostile)
New Salem (Biblical)
New Boston (England)
New Salem
New Athens (Greece)
New Berlin (Germany)
New Lenox (England)
Indiana
New Albany
New Durham
New Garden
In Iowa, there’s a New Buda (Hungary) and New Gottland (Sweden)
Let’s leave it at that. I have to confess, though, that I cheated, I used an external help for my brain… but I have read these in a book — in the early 1980s I read a book about US place names, so I had the idea that “New Something) is the most common repeating pattern, especially New Londons and Yorks — which tell about the people, where they come from.