Same process as my answer to the previous question, but adapted to place names ;)
Use the same process online name-generators use
I'm not sure of the protocol for providing answers that are pretty much just links to other answers on SE, but the answer to this worldbuilding question sounds like exactly what you're looking for.
tl;dr Define a set number of linguistic building blocks ('ne', 'rt', 's'en' etc.) and combine them using a random number generator. The set number of 'blocks' will help give your language a distinct 'sound' to it.
You can then go further by defining some grammatical rules like 'ab never follows aa' and cross out words that use that combination.
Adapting that to place names:
Layer 1: Likely name structures
One of the things places are often named after notable features of the landscape, or for notable activities. If there's a big rock nearby, a village near it will probably begin life as something along the lines of 'big rock town'. There are tons of real life examples of these, but 'Oxford' is a pretty good one. It is, quite literally, a place where people took oxen across a ford on the river Thames.
Now, place names have a lot more permanence than names for people. They tend to last and last, through invasions, and through cultural and linguistic shifts. What this means is that place names tend to be quite archaic in their descriptions, and also get corrupted over the ages.
An example of the former is the village I grew up in: 'Lingfield'. 'Ling' is an archaic word for the plant heather, so anyone around when it was named would know it was probably near a field of heather. However, anyone now would need to know that 'ling' used to mean heather.
An example of corruption over time would be 'London'. It's etymology is debated, but one theory is that it began as the Celtic 'Lundeinjon' (later Lundein), then Latin 'Londinium', then Anglo-Saxon 'Lundenwic' or 'Lunden' (near enough), to English 'London'.
So, what we need to do is arbitrarily choose some prefixes and suffixes from the same linguistic building blocks of our initial language to mean things like 'town' or 'fort' or 'river' or 'lake'. Other options would be things to do with livestock, or professions. We can then mix these up to produce some place-names that mean things like 'sheep-village' or 'mountain-fort' or 'smith-town'.
Another thing people like to do is name places after themselves. There's a whole string of 'Alexandria' variants spread across Alexander the Great's conquests. You can also throw in some of your people-names form earlier to make things like 'Tony's-fort', or 'Bill's-village'. People also like to name places after popular religious or cultural figures, so choosing a few of them would be good for variety too ('Wednesfield' is derived from 'Woden's field', for instance).
Layer 2: Invasion and linguistic shifts
When places are invaded, or there are other large-scale cultural and/or linguistic shifts, some of the building blocks of place names will change. People with these differing languages will sometimes found new towns, or rename old ones (sometimes gradually, sometimes immediately). Sometimes places will get renamed wholesale (e.g. Constantinople to Istanbul). Sometimes languages are close enough that they blend together.
You can use this to showcase some of the history of your world, and give it a feeling of depth and subtle variations in local culture.
As an example, the suffix '-ton' indicates a farmstead in Old English. The suffix '-by' means a similar thing in Old Norse. Starting from the 800s BC, Scandinavians began invading and settling in the east midlands of Britain, creating an area called the Danelaw. We can track its extent by the prevalence of place-names ending in '-by' as opposed to ones ending in '-ton'.
To include this, we need a second language with at least some different linguistic building blocks, and again come up with a list of prefixes and suffixes. for areas that have long been under the sway of this new culture you can outright replace place-names with their foreign equivalents. As their influence wanes towards the frontiers, you can start blending suffixes and prefixes from language A and language B.
Start mapping out the history of multiple invasions and you can devise a very complex and quite realistic map of place-names for your culture.
Layer 3: Linguistic shifts
If you really want to go down the rabbit hole you can start developing how your linguistic blocks changed over time, and use that to corrupt your place-names. However, you can probably avoid this entirely if you don't use your new language in regular speech. If you do, perhaps choosing slightly altered vowel sounds for your place-names compared to their modern language equivalent would do.