I didn't experience serious bullying, but I was an extremely nerdy, introverted girl who was younger than the other people in my grade and therefore often excluded from popular-kid stuff. The thing that saved me was reading - I went through book after book about kids like me (or adults like me) who weren't part of the high-status in-group, but who had something much more important going on that wasn't immediately obvious. A Wrinkle in Time is great for this, if your daughter is into fantasy books - it's pretty old but it's classic. I identified more with characters from books than I did with the grade eight social dramas.
I also found that it gets easier. By the time I was nearing the end of high school, I had a tight group of equally bookish, nerdy friends and we're still friends a couple of decades later. It takes a while to find "your people".
I'm usually not that keen on tweens spending lots of time on the computer, but perhaps on the internet your daughter could connect with other kids in other places who share her interests (with parental controls on her usage, of course)?
I'm not sure there's much point in talking to the other kid's mom, unless you know her pretty well - too much potential for mom drama to add to the kid drama. (On the other hand, if I found out that my daughter was "voting out" other kids from lunch tables, she'd be grounded for, well, the rest of her life).