What can I do with my kid in New York at night? Any indoors activities (because it's raining)?
You could try the American Museum of Natural History and the Hayden Planetarium (which is inside the museum). They often have evening shows, etc by age group. If not, Central Park - SummerStage has shows at night and also free shows on the weekends.
Mentioned in this answer:
Head to Serendipity 3 for food then over to Rockefeller Center.