Dining along the lake - not in Chicago
Years ago I was a guest of someone at the restaurant Waukegan Yacht Club. I've no idea if the restaurant is open to the public. And, I don't recall anything memorable about the food (although, if it was awful, I probably would remember). It was right on the water. You might give them a call if you have your heart set on a lakeside lunch on the North Shore.
I can't think of a single place. My best suggestion would be to make a picnic either at the beach, or (if you are not into eating that close to the sand) in one of the parks adjacent to the beach. There is a beautiful park in Wilmette near the Bahai Temple or there is a park area adjacent to the Glencoe beach. The Glencoe park is nice because it is above the beach, and you can look out on the water.
If you go to Glencoe, you can get yummy picnic food at Foodstuffs or Meg's Cafe, both are on Park Ave., less than a mile from the beach.