Lunch near Lake Michigan bike trail
What day of the week? And are you looking for a nicer sit-down lunch, or a "quick/cheap eats" type lunch, or a carry-out to take with you for a picnic?
You may want to check out the suggestions in this topic:
picnic food for Lincoln Park? - www.chow.com/topics/790107
For example, if you're leaving from Bobby's, you can get a carry-out lunch/picnic at Fox & Obel. If you want something in the vicinity of the north end of Lincoln Park, there's a great selection of cheeses and sandwiches at the Lakeview location of Pastoral.
And if you want a nicer sit-down meal, the choices vary; some places are open during the week, some are only open for brunch on Sundays, the Green City Market is Wednesdays/Saturdays, etc.
Fox & Obel Food Market
401 E Illinois St, Chicago, IL 60611
Pastoral Artisan Cheese, Bread & Wine
2945 N Broadway, Chicago, IL