I agree that Funtime Junction is a good choice. She will love the moon bounce and the indoor gym. There's an arcade too. However, I think maybe Imagine That! children's museum in Florham Park is a better choice for a special day. Depending on where in Clifton you live, it's roughly 30 minutes from you, and has lots and lots to do. A pretend ballet studio, pretend supermarket, pirate ships, planes, trains, etc. to climb up, slide down, and explore, a music room, a playhouse, post office, dentist office, etc. Cheaper, better kept up, and better laid out that the children's museum in Paramus. Both Funtime and Imagine That have a cafeteria for food; Imagine That also allows you to bring your own food.
My advice - call both places to find out if they are expecting any large groups of school age kids - they can spoil the fun a bit for a 3 yo.
Happy Birthday to your "baby" !!!