I assume that there will not be a lot of 1 year-olds in attendance. We had a really great, extremely reasonable dinner at The Little Mexican Cafe in New Rochelle on Saturday. I wouldn't think that anyone would care if the baby got cranky or if you brought in your own cake. Ciao is a good suggestion, too. The pizza is probably better than much that's on the menu there, although I happen to know that the meat is supplied by Biancardi's on Arthur Avenue, thus pretty darn good. Try the veal chop...
Almost every one of my friends has had there kids first birthdays at Ciao's in Eastchester. Personally I don't love the food, but you get the upstairs or downstairs to yourself. The service has always been exceptional. They bring out an antipasta, thin crust pizzas which kids love, then a pasta course, and usually three choices for entrees. Last time I was there, it was a chicken, veal, and beef. Although I believe you can choose a fish or another pasta course. Don't quote me on this, but I believe it's about $35 a person