I work on REI's Midwest Experiences Team and polled my colleagues in Chicago for ideas. I'm excited to help you find the right path (pardon the pun!) -- Here are a few recommendations for your family to consider:
Indiana Dunes National Park - The Glenwood Dunes Trail System connects to a series of old roads (visible on Google Maps) that lead to the Dunbar group campsite at Dunbar Avenue Beach. It's approximately 5 miles of hiking one-way. Fall is a great time to visit and there are usually fewer visitors mid-week.
Chicago's Outerbelt Trail - Suggested overnight route from Camp Sullivan to Camp Bullfrog. Check out this recent how-to article from Thrillest for more information. Very close to home, this 5-star trail is an absolute must for any Chicago local.
Hoosier National Forest in central Indiana (though, much further away) has several nice options varying in length and difficulty through scenic hardwood forest and varied terrain with views of nearby Monroe Lake. Primative campsites and dispersed camping available. Consider the Sycamore Trail Loop or Pate Trail Loop.
We noted several other State Parks or Forests with great hiking trails and more of a traditional style campground: Sand Ridge State Forest, Starved Rock State Park, Matthiessen State Park in Illinois and Maquoketa Caves State Park in Iowa.
I hope this is helpful. Happy trails to you!