The first is something a lot of people have done anyway for some time, use a pseudonym. Tons of people use fake names on Facebook, whether it’s to be clever or avoid unwanted friend requests, it works. However, using a fake name is against Facebook’s terms of service and people who chose this route do so at their own risk. Although one can argue that Facebook has no time to filter fake names much like they had no time to deal with illegal contests (which is why those rules changed). The reality is there is a lot really nasty stuff going on on Facebook that we don’t generally see and they tend to prioritize finding and reporting that kind of thing to authorities more than worry about some jackass in Minnesota who changed his name to ‘Turd Ferguson.’
If you don’t want to break the rules (or to compliment your rule breaking), you have the option of not using a photo of yourself as a profile picture. This is pretty much a no brainer, chances are someone has the same name as you on Facebook (there’s over a billion people on Facebook), having a random pic can be just the deterrent you need to keep your stalker ex scrolling down the list.
Lastly, you should always make sure your sharing options for photos, status updates etc. are not set to public, that means anyone who finds your profile can see basically everything you’re doing without being connected to you whatsoever. You should also take a look in your privacy settings and consider enabling the option to limit the audience of past posts to friends, this ensures that anything you have posted will not be viewable by any random creeper who stumbles upon your profile some how.