It is surprisingly untrivial to copy a file's name using Web UI.
Let it be additional context menu with options, say, to either copy file name or the full path. This option can also support multiple files selection, this way multiple names would be copied at once.
Any click near the file is triggering some action, you can't just select the name using your mouse cursor.
Right now the only workaround I see is to start renaming the file, copy the old name and cancel renaming. Which is odd.