It depends on what you're looking for. Many ethnic restaurants are open for lunch and all afternoon. Many nicer non-ethnic restaurants are not open for lunch, except for those in the downtown area. And I assume that by "north of downtown" you mean north of the entire downtown area (i.e. not just the Loop, but north of say Michigan Avenue, the Gold Coast, and Old Town).
Also, wherever you decide to go, you should probably call ahead to make sure they're open. Since it's a day when most people are off work, some restaurants may not be following their normally scheduled hours (which means some could be open when they are normally closed, as well as vice versa).
For Thai food, my favorite Thai restaurant,
Thai Sookdee in Evanston, is open for lunch. It's fun and I think it's got the best Thai food in the Chicago area, particularly their tom kha gai (soup) - mmmm! I know that there are some other places on the north side that have a Thai language menu and some claim are more authentic, but I've been to those, and for the more common Thai dishes I prefer, I think Thai Sookdee does a much better job.
For Mexican food, there are a few choices open for lunch, but some of them are pushing your geographical boundaries. My favorite Mexican restaurant by far is
Flamingo's Seafood, but it's in Mount Prospect (near O'Hare), which is well west of Cicero. Sol de Mexico is a possibility; I've found that the food is okay, but I haven't been overwhelmed with it. It's right ON Cicero Ave. Short of Flamingo's, and excluding downtown places like Frontera Grill and Salpicon (Salpicon is not open for lunch, anyway), my favorite Mexican place is
Lupita's in Evanston. If you go there, I recommend the innovative daily specials, rather than the more conventional items on their regular menu.
For Indian food, the many restaurants along Devon Avenue between Western and Sacramento are always fun. Most offer a buffet deal as well as food made to order.
Mount Everest in Evanston is also good for Indian and Nepalese cuisine.
Other suggestions:
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Miramar in Highwood or
Cafe Central in Highland Park, for French bistro
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Sola in North Center, for contemporary American
- Dixie Kitchen in Evanston, for Cajun
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Prairie Moon in Evanston, for American
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Convito Cafe and Market in Wilmette, for Italian bistro
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Mitchell's Fish Market in Glenview's the Glen Town Center, for seafood
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Player's Grill in Highland Park, for fresh seafood in a sports bar atmosphere
All of the above are normally open for lunch on Fridays, but again, it's best to double-check with them to make sure they will be open the day after Thanksgiving.