Good new is, if someone were to rename their character, you won't lose them on your friends list.

I know this because a friend transferred off server (and their name changed to 'deleted') then transferred back as a different name and I didn't have to refriend them.

Bad news is that, I suppose, if you were changing your name to avoid people, they could find you pretty easily if they chose to. Your lodestone page number does not change, anything you crafted that has your name on it will update with the new name and anyone ever on your friends list will see the change (even if you have removed them, it doesn't remove your name from their list, only yours).