I am the queen of budget travel! I've done Europe and most of the US on a pittance.
Here are my suggestions:
One--reduce your amount of time you want to be there. 2 weeks is enough time to really get to hang out with your friend and spend some quality time w/ her without spending a ton of money, overstaying your welcome with her family, or possibly getting sick of each other--yes, its quite possible (ask anyone who's married! Just kidding)
Two--start looking now for cheap plane tickets, train or bus fares. Consider calling a travel agency that specializes in student travel. There are a number of them in the Chicago area--google "student travel Chicago" and you should get some results.
Three- Be prepared to rough it a little. I'm a big fan of saving money on meals by combining breakfast and lunch into one big midday meal. I'll have coffee around 8 or so after waking up (most hotels have free coffee) and then I have a big "brunch" at a diner or someplace kind of cheap so that I'm killing two birds with one stone. Plus, brunch is fun! It's nice eating pancakes at noon!
Four--Talk to your friend about some things you all can do together that are free such as boardgames, maybe doing a big puzzle together while you stay with her. Things to occupy your time so you aren't spending money you don't need to be spending. Last year, when my husband and I had just moved to Chicago, before we both found jobs and I started grad school, we were on a very strict fixed budget but were essentially on vacation in a new city--we found ALL the free things, free days at museums, coupons for restaurants, free parks, and really got to enjoy ourselves in our new town before reality kicked in and we had to start our routines. There's deals to be had if you look hard enough!
Good luck, and my suggestion for asking your mom is this: come prepared with some ideas of what things will cost, maybe even print up a "sample itinerary" from expedia or some other travel site. If she sees that you're serious and are putting effort into making the trip happen, she'll realize how much it means to you and will be more likely to say yes!