I think whether or not you know the girl or she knows you by any mutual friends is irrelevant.
The number of 100% random girls who I have added on facebook(who I have some mutual friends with but never seen or met them(or the mutual friends :p)), is incredibly high
For me it's often a hit or miss depending on if I can hit on the right vibe or not. (I suck at maintaining my state from day to day haha)
You just basically have to make them laugh, or just do something that makes you not seem creep, or attention seeking.
For facebook opens, be funny, random, relaxed(this can be seen in how you're talking), but speak with an energy.
Do not jump through her hoops. If she ask you why you messaged her, or why you're talking to her. Respond in a spontaneous way, funny way, that doesn't offend her. My advice is sort of vague, because i'm no pro at it, but this is just what my experience has been on the success I have had.
On the success I've had, once I get the girl in my funny and random vibe, I can usually drive it towards sex talk, her wanting to fuck, her wanting to hang out, her wanting to smoke, etc, what ever. Just make sure you throw a bit of game in there, subtle DHV's. Try to express personality DHVs in every response; funniness, confidence, non-reactance, etc.
Btw, on facebook, always say less than necessarily. Do not type excessively long responses. Keep them to concise and short sentences. If you seem you are putting too much effort, and wanting her to reply too badly, this will show. Don't send paragraph responses, or ones that look like paragraphs. Law of Power #4) Always say less than necessary. If for some reason she doesn't respond in 5-10 minutes, engage in the random/funny responses, but link it to her ignoring you in a SUBTLE way. Don't point out she's ignoring you, but subtle imply you wont stand for it in a funny cocky way(I don't have an example of this, sorry, it just comes to me depending on the context of the conversation). - If she ignores you, then make sure whatever you reply back with is funy, cocky, calling her out SUBTLY, AND relates somehow to your previous message exchanges.
I once opened a girl with talk about Purple Elephants and Flying Tigers. It continued from there, bridged into her talking about favorite colors, etc etc. Lead to her talking about sex, then eventually how she wanted to get me in her sex dungeon. Lol

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I figure the main theme in facebook pick-ups is that they have to have very little logic behind them. Focus most on the personality and vibe that you are giving off. Remember, facebook isn't like a club. Girls go to clubs to socialize etc. Facebook is it's own field, and when you understand it, it becomes very easy. People go on facebook to socialize with friends and spend their free time and or waste time, them typing to you doesn't take them away from their friends conversations. Really a lot of people on facebook are there for no reason and if you can be entertaining without making a complete fool of yourself then you can grab any girls attention. So you can pretty much talk about ANYTHING, as long as you manage to open her in an effective way that gets her into your VIBE.