Actually, although so much of @LuckyGuy‘s advice is spot on, since neither you nor your daughter understands this stuff very well, I would not recommend the parts about his responses to “play with the guy.” That would let him know that you know about his activity. Better he should think that she has quit Facebook (which I would also not necessarily recommend) than to let him know now that “she knows”. That would alert him that he has to improve his methods, and make him smarter, drive him underground, and let him know that he can no longer tip his hand. Since he’s already ahead of you in this game, that would cause him to up his game so that he would stay ahead. I think if he reconsiders his advice, even @LuckyGuy might agree with this: You aren’t ready to play his game yet in a way that @LuckyGuy would be.
Better for him to think that your daughter has just given up on social networking.
In the meantime, as both @LuckyGuy and @ETpro have suggested, you need to use passwords that are much more (apparently) randomized and secure. Longer. More special characters. Mix in numerals, capital letters, punctuation marks (if the system allows that). They don’t have to be entirely random, but number-for-letter substitutions can help you with mnemonics, such as
YuZel0tz0f$pec1a11charz
It’s not hard to remember “use lots of special chars”, is it? Then just remember how to write it differently, in this way. She can remember that, surely (or something like it) and let the numerals and special characters make the password more complex. And change passwords from time to time. Don’t use the same ones over and over. Don’t type them onto screens where anyone might be able to “shoulder surf” and pick up the keystrokes.
Another thing that you’ll have to do, as well as her, of course, is be very leery of social engineering. He doesn’t have to know all of her habits and all of her messages… if he has someone else pass those on to him, say, someone who still trusts him and thinks that he and your daughter should still be a couple, for example. So your daughter has to be aware of who her friends really are, and not be so open with all of them all of the time. In addition, be careful of who is admitted to “friend” status on Facebook and other networks. It’s easy, when we’ve lived open lives with good friends for a long time, to accept every friend-of-a-friend request when we don’t know the person at first hand – or we think that the person who is sending us a request is really who we think it is.
You guys need to think about security now in ways that you never did before.