Being from the IL and having visited la Suisse et la sud de France. I find that mid-western beef and pork is by far the best in the world. The only thing better is Kobe, but thats more exotic imported cuisine than typical Japanese. But I digress. I think America also invented BBQ. I know Chi-town also has good BBQ places. Can you ship frozen food?
The only other food thing Chicago does well is Chicago style hot dogs and Chicago style pizza. Neither package well.
Another thing that is distinctly American is our great variety in cultural foods and icons. I can go find Mexican, Chinese, Japanese, Lebanese, African, Brazilian, and European foods and cultures very easily. I remember that is something I missed a great deal while in Europe.
Chicago is also famous for its downtown architecture. There ought to be several really good books that would explain the history and show in more detail the beauty of the buildings that were created. It would be a good compliment to a photo album that others suggested, which is a really good idea.
I believe blues and jazz are popular in France. Chicago and Illinois has rich history there, as well.