Well everyones recommended albums so here are some individual songs
Sugar (THIS IS A FUCKING MUST)
Revenga
ieaiaio
Cigaro
Toxicity
Holy Mountains
Glad you are checking out this band they are fucking great.
Well here's the issue with your list of songs.
She's trying to get into SOAD, so we want to start her off with the non "weird" songs they have. It might be too much at once if she starts off straight away with songs like "Sugar", "Cigaro", "Violent Pornography" and such.
If she starts off listening to Toxicity (the album), it's a good start because it's a bit more tame than the other albums. It's got more of a rock-edge to it on most of the songs, while still retaining the typical SOAD oddity (albeit in a slower, more introductory way).
So she should start, no question, with Toxicity. Up next, either Steal This Album! or Mezmerize, and by then she'll be used to it all, and can move on to Hypnotize and then self-titled.
So in reference to your list, I believe Sugar and Cigaro to be a bit much to start off with. However, Revenga and I-E-A-E-A-I-O are great songs for her to listen to, because they're fairly clean, typical System sound, and extremely catchy.
Toxicity is just an absolutely mind-blastingly awesome song, so it's a safe bet that she'll like it.
And finally, Holy Mountains. What a great song. This is pure System Imo, and just all around awesome. I feel like this song doesn't get enough praise (at least from what discussions I've had about SOAD). She should definitely listen to this.