We had a Van in South Africa for these 5 or 6 weeks our family (we flew forth and back depending on who gotten when vacation, just spent xmas together) was there. Just called some agencys and looked for the best rate and then took that one.
Usually rental agencys are pretty good if they know you will have the car a few weeks, and there are agencys who are specialized in dealing with long term rentals. What is also possible is that you ask a few car dealerships. They sometimes have cars standing around where a few more thousend miles dont matter and that they rent out...longterm might also get you a good rate.
Do you plan on returning to Chicago in the middle of your trip? Car rentals become more complex when they are longer than 2 weeks: you usually can't take advantage of the promo rates or use discount/upgrade coupons on longer rentals. Therefore, if you have the opportunity to switch out cars, you'll save some money. (Example: quick search on ORD yields monthly rates for a full-size at $900+, two-week rate is around $300. And that's before applying any discounts/upgrades you might have.)
If you can do the switching to keep these rentals in the 2-week range, then it becomes a normal exercise of finding your best rate across all companies (including airport and off-airport locations), and then letting Priceline try to beat it.
When comparing off-airport vs. airport rental costs, make sure to consider total costs including taxes and fees.
I did the mini lease for months at a time. You have to take the car back once per 30 day period. It is a good deal, costs more than owning a car but less than renting. We paid $800 or $900 per month that included everything for a nice car, I think a Grand Prix. No maintenance, just gas, everything else is their problem.
Trying not to believe what I think...AND...
Remember: I don't know what I'm talking about!