What is the significance that it is the woman who identitifes that the hills look like white elephants?
This is subject to your own interpretation. I’ll offer a couple ideas. This is a selfish, shallow man trying to convince the girl to have an abortion; which we understand without either of them ever mentioning the word ‘abortion.’ So, considering the title, as it is spoken of in the story, try to see how it is saying something else – just as the dialogue gives us knowledge indirectly.
The man is not selfish because he wants her to have an abortion. He is selfish because he doesn’t want this pregnancy to interrupt his life, which seems to be nothing more than traveling with this girl (and perhaps others) looking at things and ‘trying new drinks.’ The girl, on the other hand, is at least willing to consider having the baby and maybe even starting a new life, compromising. She’s willing to do what he is not: imagine a future that will be significantly different from their current lifestyle. She notes the hills look like white elephants. He does not see it, implying that he is incapable or unwilling to imagine how things could be different. The white hills also symbolically represent a pregnant stomach which could be an intentional symbol or not. If so, he is also refusing to acknowledge the possibility of the pregnancy. In general, the girl’s creative perception and the man’s lack of one show that she is open to new ideas.