Chicago covers a LOT of real estate and, as sunbrace noted you don't say where you are meeting your granddaughter. Guessing that you will be in the center business/tourist areas here are some suggestions:
For French, you might like Bistro Margot in Old Town or Bistro Zinc on the Gold Coast or Henri (not really a bistro) in the Loop.
For Italian, I love Bar Toma. It's pretty authentically Italian, but not exactly "old school" Italian as we know it in this country. You might really like Francesca's on Chestnut or Quartino's, Carmine's or Club Lago.
Blackbird does have a very affordable lunch special, but I'm not sure if it is open on Mondays.
Where are you meeting her? If it's downtown, you can try Rosebud, Bar Toma or maybe Cafe Spiaggia for Italian. Blackbird in the West Loop is not quite French. It serves modern American food but is very delicious, especially if they still offer their lunch prix fixe. I believe it was in the 22 to 25 dollar range.