Why was the red priestess staring at Tyrion?

No, she doesn't appear in the books as far as i recall. But i remember the casting call made a huge thing out of her being cast, and how much of a "prominent" role she would have in the new season...We'll see as time goes on what will happen to her (and to Varys, for that matter- Tyrion just left him alone in that whorehouse).

I don't understand why she stared at Tyrion but she does appear in the books claiming Daenerys to be Azor Ahai. This is significant as Daenerys fits all Azor Ahai's prophecies
1. Born of salt and smoke: She was born on Dragonstone, an island in the ocean (salt) with a bulkan (smoke)
2. Wake Dragons From Stone: She turned her petrified dragon eggs (which were stone) into live dragons