I know the name gotham refers to new york, but it predates the comics book creation for the batman universe. The oldest reference that I am personally aware of is the New York gothams in 1883 which was the first season for the baseball team that would eventually become the San Francisco Giants. So where does the name come to? And how was it created and or choosen?
The place name is old english. The etymology is simple: gat (goat) ham (home), so a homestead or settlement where goats were kept. It's not a common place name but there is still at least one Gotham in England.
This article from New York Public Library talks more about the associations of the name, how it came to be applied to an area of New York, and some further reading.