I think your itinerary is fine although I'm not sure your MIL will really want to stay too long at the City Museum. Your daughter, however, will love it.

A couple of good breakfast/lunch/brunch ideas downtown

Rooster http://roosterstl.com/

London Tea Room http://thelondontearoom.com/

Rooster can get very busy on weekends, just so you are forewarned.

London Tea Room is not far from City Museum. They have nice quiches and grilled sandwiches. They have Cream Tea anytime (Tea, scone, clotted cream and jam) as well as an expensive, by reservation only, afternoon tea. Might be a nice post City Museum activity.

I'm not a big fan of most of the restaurants on the Hill. The best Italian almost on the hill is Trattoria Marcella http://www.trattoriamarcella.com/home.html Reservations required.

If you must eat on the Hill, I think Gian-Tony's might be an ok choice. Fairly casual, old-timey St. Louis Italian, not too expensive. I haven't eaten here in years, and the reviews I just read online are mixed...
http://www.gian-tonys.com/

If you end up in Forest Park for the Zoo, Science Center, History Museum or Art Museum a good choice for lunch would either be the Boathouse in the Park itself http://www.boathouseforestpark.com/

Or head to University City and the Loop area. Blueberry Hill would be my recommendation, your daughter will think it's fun. You might too. They have good burgers and my fave fried zucchini strips http://www.blueberryhill.com/

http://www.stlzoo.org/
http://www.slsc.org/
http://www.mohistory.org/
http://slam.org/

One of the best places to search for St. Louis Restaurants is Sauce Magazine. http://www.saucemagazine.com/

Any other type of food you're interested it? There are plenty of restaurants downtown, not far from your hotel, but I'm not sure how upscale you are looking for. I would stay away from Laclede's Landing as an area to eat.

We have had beautiful weather for weeks, but it is supposed to rain a lot this weekend. But there should be enough inside activities.

We liked the Drury at the arch. Do they still offer the complimentary breakfast and a hosted cocktail hour in the evenings?

Duff's restaurant, in the Central West End, is a class act, in a great neighborhood and with a creative daily menu.

A big Yes, also, to Blueberry Hill. If you are lucky, Chuck Berry will be performing, too!