Here's the situation guys. I'm 19, because I know sometimes age has a lot to do with it. She appeared to be a similar age. HB9 (not sure if this matters), she looked a mix of hispanic and something else, nice long hair, about 5'3, and a NICE body. I'm a good looking guy as well, I'm told that by everybody. I participated in a mock trial today as an expert witness (anti-gang expert, LT. for the police dpt, and I knew my shit. DHV?), and I noticed her in the jury. I kept giving her glances to see if I could get some eye contact and on two occasions she was looking in my direction. I'm not sure if she was looking directly at me or not. I planned to open her when the trial was over.

There was only one problem. The trial ended, I got into a conversation with the prosecutor. When our conversation was over, I looked around and she was nowhere to be found. Went outside of the court room to check near the elevators and she was already gone. I couldn't be fucked to go run after her. She was the juror that read the verdict and she had to state her name. This is how I know I can open her on Facebook, but is it a good idea? How would I approach this without seeming like a stalker? When should I add her?

I was thinking I could wait a couple of days and then send her a friend request telling her that she popped up on "People You May Know" and I thought she looked familiar, but I'm sure this is used a lot and I'm not sure it will start the type of conversation that I want. Are there any good Facebook openers for this situation? I have no experience with online game so any tips on how to transition to getting the number from Facebook would be golden as well.

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Uhm, I doubt you can pull this off.
Will be creepy etc.
Just try adding her and tell u what u told us, you will risk coming off across as creepy but its not like you will ever see her again. Also, noone will find out, she wont tell her friends about you, so you got nothing to lose.

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We are what we repeatedly do.
Excellence then, is not an act, but a habit.

Have you considered other options?

I actually disagree with this. I was chatting to a hot girl at the gym a while ago and I didn't ask for her number because I thought I'd see her there again. Unfortunately haven't seen her since. I found a receipt the other day and noticed it had her full name on it. The thought came to mind that I could look her up on Facebook.

I haven't though, and I refuse to. She didn't even tell me her name, it just wasn't part of the conversation, so there's no way to casually act like "Oh I just noticed you on here".

If you're saying that you didn't even interact with this woman directly and you're then looking her up based on just overhearing her name then that's even worse.

You're only 19 dude so I mean no criticism here at all. I would have been asking similar stuff at your age. But you need to start training yourself into good habits now, and pursuing girls online based on very tentative leads like this would be a very bad habit to get into.