Does the hash of a file change if the filename changes?

The hash of a file is the hash of its contents. Metadata such as the file name, timestamps, permissions, etc. have no influence on the hash.

Assuming a non-broken cryptographic hash, two files have the same hash if and only if they have the same contents. The most common such hashes are the SHA-2 family (SHA-256, SHA-384, SHA-512) and the SHA3 family. This does not include MD5 or SHA-1 which are broken, nor a CRC such as with cksum which is not a cryptographic hash.

Not as far as I can tell after a simple test.

$ echo some contents > testfile
$ shasum testfile 
3a2be7b07a1a19072bf54c95a8c4a3fe0cdb35d4  testfile
$ mv testfile newfile
$ shasum newfile 
3a2be7b07a1a19072bf54c95a8c4a3fe0cdb35d4  newfile