Can check: My Documents\My Games\Final Fantasy XIV - A Realm Reborn\FFXIV.cfg
Right click that file -> Properties -> (On the general tab) -> At the bottom is Attributes -> Make sure Read-only is unchecked.
If it was checked, that's probably your problem.
If it wasn't .... Sounds like a problem with "Windows UAC" and basically what's happening is the game doesn't have privilege to modify the file.
You could try right-clicking your shortcut to FF14 and 'Run As Administrator' but this may only apply to the patcher and not the game running afterward.
You could figure out what program actually runs while playing FF14 and set it to always run as administrator.
You could try turning off the 'Windows UAC', go into the game and make the changes, then exit the game and turn the 'Windows UAC' back on. (might have other issues in the future)

Answering this while they're doing the 2.1 patch, which may fix your issue and you not having to do anything.
Lrocenahs: ****Be very careful when you manually edit the file. I recommend you view it full screen or at least make sure no line is affected by word wrap. Making a backup copy of the file if you screw things up
wouldn't hurt****
If nothing else, you should be able to edit "My Documents\My Games\Final Fantasy XIV - A Realm Reborn\FFXIV.cfg", scroll down to <GamePad Settings>, change AlwaysInput ...the 0 to a 1
so it'll look like
AlwaysInput 1
If notepad has trouble saving, you either have the file settings set to Read-Only, or you need to run notepad as administrator, then inside of notepad(run as administrator) open the file and make the changes.