It's called a "stay of mandate" pending a petition for cert and such stays are common in high profile cases such as this one. All Illinois need show under Rule 41 FRAP is that the petition for cert would present a "substantial question" and there is "good cause for a stay." They really don't need it to merely file a petition, as they already got a 180 stay of mandate, which is way more than the 90 days for seeking cert. Of course, a stay of mandate would operate to stay the case until the SCT issues an order on cert. If cert is granted, then the stay continues in place. If cert is denied, then the mandate issues "immediately." They have until midnight tonight (Jan 8, 2013), Chicago time, to file any rehearing petition.
PS: Al, I think the 7th Circuit can take as much time as they want to decide whether to grant rehearing. Nothing in the rules impose any 10 day limit for consideration of such a petition.