Why Fiamma? (Spoilers)

1. Mentally unstable to begin with
2. She loved that Fiamma had traveled and experienced so much culture. Fiamma was living the life Ms. G wanted to live, and told the girls she'd lived, but never had.
3. Fiamma was of a different culture. Mysterious, different, beautiful, etc.
4. Sometimes there is no reason at all. People become obsessed over specs of dust. When you're insane, you're insane. No rhyme or reason sometimes.

I thought it was because she wasn't fawning all over her over. She was quiet and mysterious. She was also living the life that Miss G fantasized about.

We live by the Sun, we feel by the Moon

Yeah, I'm inclined to agree that there probably wasn't any rationality to her obsession. But Fiamma was quite different from the other girls in that she was a foreigner, well-read, seemed to have had more romantic/sexual experience than the other girls, but most importantly I think her resistance and being unimpressed by Miss G, unlike the other girls, attracted G to her. She had the attention of the other girls, but not Fiamma, and she became desperate for it.

"A half-finished book is, after all, a half-finished love affair."