My favorite place for gyros is not in Naperville, but on the West Side of Aurora. It would be a field trip for you, but then you could do this on the way back from Woodman's or tie it into a visit to the Wurst Kitchen.
West Side Gyros is on Illinois Street -- just carryout. The gyros are packed with meat, onions, and tomatoes, and come with a plastic cup of cucumber sauce. The pita is nice and warm, and is nearly too much to eat for me. Nearly. Because West Side Gyros has the best french fries around. Seriously. They're crinkle cut, and hot, salty, and just greasy enough. I live on the east side of Aurora, and the fries are always gone before I get home.
The hamburgers are good as well, and they serve traditional Chicago dogs (Vienna All Beef, of course). They also have an Italian beef, but because I love the gyros so much, I haven't tried it. They also have great milkshakes.
The gyros come as either a sandwich or a plate, which includes more meat and the french fries. The plate is really a lot of food. Yummmmm. And the people working the counter are super nice. No matter what or how much you order, they always respond that it will be ready within 10 minutes, and by the time you get there to pick it up, it's ready!
West Side Gyros
930 West Illinois
630-553-7272
Gemato's Grill has been discussed on another thread, but mostly in connection with its grilled meats. I still drive way across town to West Side Gyros when I want a gyro, even tho Gemato's is a lot closer.
http://www.gematosgrill.com/
Gemato's Grill (near the Jewel on car-dealership row)
1566 W Ogden, Naperville
630-778-9990