One of the great advantages of origami kits is that they're all-in-one. You don't have to shop for a book and then shop for origami paper: a kit has it all nicely packaged, ready to go.
Origami Fun Kit for Beginners is a compilation kit with 3 small origami books and 96 sheets of origami paper. The books are "Birds in Origami", "Favorite Animals in Origami", and "Easy Origami": a total of 55 projects. Origami paper comes in solid colors, multicolored, and metallic. It's a great kit for a beginners. Approx $10 at amazon.com
Origami on the Go by Margaret Van Sicklen is a travel origami kit. Inside the suitcase-like box, you will find a 144-page paperback book with instructions to 40 origami projects; 100 sheets of origami paper; and stickers. Included is travel-wise information ($US --> Euros, Seven Natural Wonders, key words in different languages). The airplanes, jumping frogs, and finger puppets will keep your kids occupied during travel. Approx $10 at amazon/com
There are many many origami kits available on the market including: "Ultimate Origami Kit", or "Origami Spectacular Kit!", and the "Complete Origami Kit for Children". Some kits have a theme such as the "butterfly kit", the mythical creature kit, the paper airplane kit, and even a Christmas origami kit". Almost all kits are a combination of an instructional origami book and appropriate-patterned origami paper. See entire list here. See entire list here.
Specialized Origami Kits
Paper Doll Origami Kit by Aitoh. A specialized kit for those who love classic origami paper dolls. Kit includes instructions, origami paper, templates for heads, and special paper for hair. Enough material to make 8 or 12 dolls; glue and scissors required (not included). There are 2 versions: one is approx $10 while the other is $13; available at amazon.com.
Thousand Cranes Kit by Kidstoyo. In Japanese culture, it is believed that your wish will come true if you folded 1000 paper cranes. Most notable is the story of Sadako Sasaki and the 1000 cranes. This kit contains 1000 sheets of multi-colored paper (3" square), instructions (in Japanese), string to thread the cranes together, and beads to keep the cranes in place. Approx $15 from amazon.com.