Emily Oaks Nature Center is small but nice for a quick walk in the woods not too far from home (15 minutes). North Park Nature center is about the same. Nice, but still pretty urban.
River Trail Nature Center up on Milwaukee just north of Lake is nice, but would stretch your hour limit. They have a nice interpretive center with animals and a dopey squirrel that I adore.
My absolute favorite nature preserve near the city (again, it would be pushing an hour) is Ryerson conservation area and the trail of woods south of it (Potawatami woods, Allison woods). That's up near 94 and Deerfield.
Harms Woods (Harms and Golf) is okay if you can get past the people training for the tour de france, I much prefer the above places though.
We're "walk in the woods" sorts of people so we've pretty much explored everything near the city and if you haven't yet, you need to go up to Riverwoods and check out those preserves (Ryerson and the others) because they are simply the best, most natural ones you're going to find near the city. We even saw a beaver there once.
Forest Preserve of Cook County, Willow Springs. A little past Midway airport. It's huge! About a 40 min drive depending on traffic. Or LaBagh Woods. It's not very big- good for about a 40 min if you walk slow. It's close- right down Foster before the Edens. Look for signs for Irene Hernandez Picnic area.
When you're looking for a little walk, close to home, try North Park Village Nature Center - Pulaski north of Bryn Mawr. The property is big enough to showcase several different landscapes / ecosystems native to our area. The trails are easy for kids, and there is a nice nature center attached.
+1 for Ira Hernandez woods. Easy bike ride from Lincoln Square, nice walk along the river, follow the old railroad tracks, or get on the bike trail. Easy to spend an hour or more without leaving the city. Foster just west of Pulaski. Tons of parking if you drive.