If one is only choosing four books, they have to carry a lot of weight. I think they should be books that most of her classmates have read, too, so that it increases the store of knowledge she has in common with them, and they should be good introductions to key issues in America that people in other countries may not quite get.
Chief among the latter is race relations. For that, I nominate Toni Morrison, Song of Solomon, Other possibilities include Ta-Nehisi Coates, Between The World And Me, or Colson Whitehead, The Underground Railroad, or Alex Haley, Roots. Not Uncle Tom’s Cabin, sorry. Huck Finn is too subtle, and requires too much context. Same with Their Eyes Were Watching God, one of my favorite books ever, but not the first book to read on this topic, especially for someone not familiar with regional dialects in America.
Vietnam still hangs over us. Practically every student has read Tim O’Brien, The Things They Carried. So should she. And/or maybe Graham Greene, The Quiet American – although depending where she comes from that perspective may be available to her already.
How about a young-adult novel everyone has read, but that hasn’t been translated everywhere? Like Laurie Halse Anderson, Speak. That touches a lot of important bases, too.
Last, I’d look for something that gives the right feel of American triumphalism and exceptionalism. Whitman’s poetry. One of the main Willa Cather books. Maybe Little House on the Prairie. Or maybe Michael Chabon, The Amazing Adventures of Kavalier and Clay. That’s very popular, and gives a sense of aw, shucks greatness and excitement. Maybe the Steve Jobs biography.
Or maybe something that gives a sense of how the American political system works. Not sure what that is anymore, though…