It depends on what you're looking for, in terms of budget, atmosphere, etc.
Fox and Obel is an upscale gourmet grocery store with a cafe in the back, where they make salads, sandwiches, etc. The atmosphere is rudimentary coffeehouse, the food is very good.
Some of the best high-end restaurants are open for lunch, including NoMi and Seasons, and also some of the steakhouses (Capital Grille, Saloon).
For less expensive fine dining options, Bistro 110 is not too far away, for French bistro food. Other choices are Shaw's Crab House (seafood), Coco Pazzo Cafe (Italian), Gino's East (deep-dish pizza),
All of these are open for lunch and all are walking distance from the W at 644 N. Lake Shore Drive (thanks for the clarification, sorry the name didn't register).
Fox and Obel - www.fox-obel.com
NoMi - www.nomirestaurant.com
Seasons - http://chicago.fourseasons.com/chicagofs/dining.html
Capital Grille - www.thecapitalgrille.com
Saloon - www.saloonsteakhouse.com
Bistro 110 - www.levyrestaurants.com/Levy/DiningExperiences/Restaurants/Group4/Bistro+110.htm
Shaw's Crab House - www.shawscrabhouse.com
Coco Pazzo Cafe - www.cocopazzocafe.com
Gino's East - www.ginoseast.com