Voila, here is the annotated list of all local UWS kid-friendly places we like, as collected by Karen Davidson-Weitz.  Thanks, Karen!

Isabella's (great for brunch)
Fred's (also good for brunch)
Sarabeth's* (best during the week when less crowded)
Sambuca*
Mana (Amsterdam/91st-92nd; organic vegetarian and fish)
La Grolla (Amsterdam/81st; pizza, pasta and grown up food)
Gabriella's* (Columbus/93rd)
Artie's* (Broadway/82nd; go early, it can get crowded)
Any pizza place with a doorway wide enough for your stroller
Ollie's*
Maryanns* (Broadway in low 90's)
Fairway (upstairs cafe and steakhouse) *
Columbus Bakery
French Roast
EJ's (Amsterdam/81st)
Patsy's Pizza*
Pain Quotidien (Columbus/72nd)
Louie's (Amsterdam/81)
City Grill
Fiorello's
Vynl
Pizzabola
Aqua (good outside)
Le Monde
Popover Cafe (Amsterdam/86th)
Nikkos'* (Broadway/77th)
Boat Basin Cafe*
Café Viva (Broadway/97th; healthy pizza and Italian dishes)

Restaurants that seem to be a little more accommodating to children/babies:
Gabriella's is big and loud, perfect for muffling infant screams, nice outdoor area that is semi-air conditioned.
Henry's or Henry's Ditto (Broadway/105th) is very child-friendly, they handle the stroller for you.
Metro Diner has fabulous service for parents with kids, special seats that attach to the booths.
Nice Matin is a little more upscale, so earlier is better.
Homer's Donut Shop is geared towards kids.
Cafe d'Soleil (Broadway/104th) is extremely accommodating to babies.
City Diner* (Broadway/90th) has lots of windows for kids to look out, room for stroller, quick service.
P.D. O'Hurley's (West 72nd) Plenty of room in back area, baby-friendly food.

* More than one mom recommended

Paris's favorite comment from another mom:
"You can take NYC kids anywhere, pretty much, as long as you are prepared - i.e., don't go when they are starving or tired. Bring things for them to play with (crayons) and if they act up, you need to leave - just get your order to go so the kids learn what what eating out is (and the restaurant invites you back!)."