I really love the name
Amethyst, I think I would've used it as a stage name if I were her. However, a part of being famous is creating your own image. Maybe
Amethyst didn't fit the vibe she was going for. Personally, I think
Amethyst sounds like an electronic pop DJ, or a synthetic pop group, not a rapper. But then again, rappers have all types of names. There are a lot of options. I guess
Iggy Azalea just clicked for her. I think it suits her, anyway.