In Illinois, this is called post-closing possession. It's not common, but it's not unheard of either. It often comes into play when a seller has to sell their place, purchase a new place, and the purchase gets pushed back. If you are going to allow this, you will want to enter into a written post-closing possession agreement. Your attorney can draft this. When we do this for clients, we charge 1/30th of the buyer's principal, interest, taxes, insurance, and assessments per day as rent, collected at closing. At the end of the two weeks, I would suggest a daily rate of $500. This would deter your "tenant" from staying longer than 2 weeks. Finally, an escrow should be held just in case your "tenant" damages the property before moing out.
Attorney Ranj Mohip is a Chicago real estate attorney. The information in this answer is general information and is not intended as legal advice. Further, answering this question or otherwise contributing as a member of Trulia.com does not create an attorney-client relationship. Remember--consult the best real estate attorney in Chicago or in your respective area. Contact us at http://www.ranjmohip.com for more information.