It's a good idea to put together your own kit and don't purchase a pre-made adult-sized one that is filled with tiny needles and skinny thread that your child will have trouble with. Trust me, I've been through that frustration!
If you have a copy of Sewing School, there is a wonderful supply list in the introduction, but if you're waiting for the book to come in, this list will get you started.
chenille 22 needles
LoRan Needle Threader
straight pins with large, round heads
good pair of scissors just for sewing
Fabric and other supplies:
fat quarters of cotton fabric
notions like buttons, ribbons, elastic, trim
*Pretty much all of these items may be found at your local fabric/craft supply store. For ease, I linked to Amazon above so that you may get a visual. For more information about the various supplies, please see the "In the Child's Sewing Basket" located in the sidebar.
As for storage space, be creative! There are lots of cute containers out there on the market or you could use an old purse, fishing tackle box, or even a tool box. The important thing is that the container is big enough to fit all of the supplies (not including fabric) that your child will need for sewing. The more organized, the more successful their sewing experience will be.
If you live in the Memphis, TN area, I have some official Sewing School kits that I've put together available for sale at The Cotton Museum, The Trolley Stop Market, and Sew Memphis. You can even get it online at the The Cotton Museum!