The only problem with the recommendation of the Plaza is that it doesn't have a sit-down restaurant. The Palm Court serves afternoon tea, but not lunch. There's a casual food hall and plenty of bars, but no restaurant.

Nougatine would be a better choice, though I don't find it particularly kid-friendly.

I like the Mandarin Oriental idea. It's a fancy space, and we certainly found the service to be excellent. (Not sure where the Christmas tree idea came in, this birthday weekend may or may not be any time around the holidays.)

yeah, that's true but the Palm Court serves evening "snack fare" that is what most people would call dinner. Such as filet mignon, pan seared salmon or duck breast, as well as some sandwiches and salads.

Christina, you're making me hungry

Red Eye Grill

The Mandarin is very pretty and there are a lot of kids at the hotel but I can't vouch for the food. If you really want to be in Times Square, see Lattanzi. Not super fancy (white tablecloths and nice but not upscale, close to the theaters, consistently good, and they do half portions of pasta for the kids). Very close to Times Square (walkable but not in it) are Gabriel Kreuther and Le Bernadin. We love Maialino too but that is not a convenient location for you and is also not super fancy. Broadly speaking, you will do better food-wise if you venture out of Times Square. Somewhat upscale and easy to get to Times Square but require transportation would be: Dovetail (I've seen kids there but not many), maybe Milling Room (less upscale but kid-friendly and consistently good). Your daughter will go bonkers for high tea at the Plaza. Maybe you could do that earlier in the afternoon or the following day.