It is simply not acceptible to have to put up with a Facebook cyberbully. When someone is behaving in this manner they are causing problems that no one should have to deal with when they are just trying to enjoy their social media experience. As such, it is important to know the proper steps to take to deal with a bully should the need ever arise.
The first thing to do is to add all of the privacy settings that you can to try to prevent this kind of thing from happening again. It will also help to cease any harassment that you are currently experiencing. This wikihow article explains how to go about doing this:
Set up a limited profile so that you can control how much people other than your accepted friends can see when they find your Facebook page. Go to the "My Privacy" page and click on "Limited Profile Settings". Be very cautious about which settings you allow to be viewable by just anybody; keep most things to confirmed friends only – list their names in the limited profiles box. If you're a parent, help your teen to work out the appropriate profile level.
This is an important first step for dealing with a Facebook bully, but it is not the only thing that should be done. It is also important that the issue be address with the bully directly. It is possible that he or she does not even realize that they are showing bullying behaviors. It is rather diffcult to understand how they could not know, but there are certain people who may feel that what they are doing is a social norm when it really is not. Simply addressing the issue with them and asking them to stop can sometimes be effective.
If your self policing strategies do not work to stop the bully, then more action may be required. In some cases you may have to report their behavior to Facebook. At least then the site can determine if the behavior is bad enough to discontinue the bully's account.
In extreme cases, if nothing else seems to be working, you may consider closing down your Facebook account yourself. Although this seems like an unfair outcome, it is better than having to continue to deal with a bully who just won't stop. 