There is many places to stay in Great Lakes, I perferred "The Navy Lodge". Although your not military, you can still stay there because your spouse, son, or daughter is in bootcamp.
To get a room you need to book well in advance. The rooms are quickly booked. So as soon as you see their graduation PIR listed, make sure you book a room. You can see the graduation schedule here >>
http://www1.netc.navy.mil/nstc/rtcgl/upcoming_grads.asp
Even though the rules say you must have your sailor there when checking in, they make an exception for Great Lakes. So you don't need them there. Also remember, in Great Lake you are allowed to use the Navy Exchange while you are there...
Their Navy Lodge website is:
http://www.navy-lodge.com/index.html
Navy Lodge
Great Lakes
Great Lakes Naval Training Center
Great Lakes, IL
60088-5000
847-689-1485 -
Fax: 847-689-1489 Amenities ...