I cannot think of good names for towns in my country in the post apocalypse. Most of the time I just call the towns “New Houston” or “New Seattle”, but I need more creative names.
What are methods to come up with unique city names?
It's hard to imagine a post-apocalyptic town that's not on the site of a pre-apocalyptic town -- good town sites tend to be good town sites, after all. Let the post- names be cut down versions of the pre- name.
We see this throughout Europe, for instance, where modern towns have cut down por mangled version of the Roman town that once stood there. Chester comes from "castra" meaning "camp", Istanbul comes from Greek which was pronounced something like "is tim bolin" -- meaning "The City" -- which is what the inhabitants of Constantinople actually called their town. Modern Cologne in Roman times was "Colonia Claudia Ara Agrippinensium" -- only the "Colonia" (meaning "colony" or "settlement" survived. This happened a lot.
You can think of city names as having three parts: an optional prefix, the name, and an optional suffix. For example:
West Darlington Grove
Frankly, the real problem is the primary. And my favorite method is to choose one word that reflects the purpose for the city in my story. For example, if it's the place my protagonist is going to hide...
Next, grab some synonyms from Thesaurus.com:
Pick a couple with a few prefixes or suffixes...
and choose or start over from there. You can work with either synonyms or antonyms, and the construction actually helps as a form of foreshadowing for your story.
I would say take in the demographics of your town and use that to name your town. For example, if your town is predominantly populated by say people with Mexican or Spanish descent (I assume your setting takes place in USA) then it could have a name like Los Vidas or say predominantly Scandinavian then it could have a name like Overleaf (from the word overleve which means survive in Danish - according to Google translate).
Basically have fun with it. Put in the root word you want to use for the town (survive in this case), figure out the predominant language used, say Danish in this case and try to figure out a homonym for the word like what I did for overleve.
Of course this is just one way. Its your world so you can name it whatever you wish. You can even invent a new language and put in prefix and postfix to identify a town, like PelBrooks where 'pel' comes from the word peligro or danger in Spanish.
I think Fallout New Vegas did a pretty good job at creating and explanation of names.
New Vegas is New because they knew what was previously in this spot and they recreated "the spirit".
Novac was set in previously unnamed motel (for us) but the "No Vacany" sign had "any" letters lost hence naming place by visible name. Goodsprings - which is named because the SPRINGS are GOOD sources of fresh water.
So - let's assume that You have Los Angeles. In the first way you would just call it New Angeles. In the second it could be called "Holwod" from destroyed "Hollywood" sing. In the third option, I dunno, BurningHills?
Now, mentioned places had very distinguish thing. People (or lack of them). People are like cockroaches. You could nuke them and they crawl out of the hole in the ground and will tell you "this was New York" and everyone still call it that. But you may have place without people when the apocalypse hit and no signs of names. So settler comes and call thaw "flat planes". because it's flat and there are planes.
It depends on the "nationality" of the area. Southern US would be more Spanish and Mexican styled names, as others have mentioned. If you wanted far north US, you could use some traditional Inuit/Eskimo names, or if you don't care for true accuracy, steal some Welsh town names which tend to sound mystic (Llandaff, Harlech, Tyla Garw, Clydach Vale).
You could also look to existing towns in the US and combine parts of their names, z.B. Asherville + Sweetwater = Asherwater, Sweetville