You can’t change your name on Facebook:
They won’t let you:
1. Delete your last name.
“Your name change request has been denied by our systems. Please check that you filled out both a first and a last name and try again.”
2. Change it to an initial:
“You must provide your full name.”
3. Or do anything else:

Now there’s really no way to prevent them from making your real name public to everyone. Except changing it to something fake that will confuse your friends. This is creepy and wrong, especially for those who may be in situations like these.
EDIT: A friend just suggested trying special characters anyway. So if you’re Jane Doe, try Jåne Døe, etc.
EDIT #2: Changing your name doesn’t help anyway, because Facebook stores your old name for search. As far as I know, there’s now no way to remove it without deleting your old profile entirely, logging in with your new fake name, then readding all your friends with a message explaining that it’s you. Of course no one new will be able to find and add you, but then again, I’ve had next to no new random friend additions over the past year.
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