score:4
I would opt for the flight that arrives the previous day. Then I would look for the cheapest hotel around the airport - make sure to look just outside the airport, and it'll be a lot cheaper than one right in the airport.
Hotels usually offer free shuttle to/from the terminal. Once you get to Chicago, call the hotel and confirm the pickup time/location - go to the hotel, relax a bit, take a shower, leave your luggage in your room - then go the city. You can take the same shuttle bus back to the terminal and then go to the city, or go straight from the hotel.
The following day, when you check out from the hotel, pretty much ALL hotels will allow you to leave your luggage there for a bit longer - closer to your flight time. Go to the city again - enjoy the whole day in the city - then come back to the airport/hotel, get your bags - and onto the bus.
You should be able to utilise the hotel shuttle bus as many times as you need. Thus, coming the day before, for the price of the hotel room, you'll have a place to relax, to wash, to leave your bags - and you'll have the whole day to enjoy the city.
Upvote:3
There is a concise guide for a short layover in Chicago, published in Forbes http://www.forbes.com/sites/andrewbender/2014/01/21/layover-guide-chicago-ohare/
The guide recommends to buy a $10 day pass on a stored-value Ventra Card. A one-way ticket is $5 from OβHare (vending machines in the station below the airport)
Great pointers can also be found here: Can I visit Chicago during a 5.5 hour layover in O'Hare?