Toon Boom Harmony: Successor to the popular Toon Boom Studio. A full-featured animation software package suitable for ambitious teens looking to make high-quality 2D animation. This is professional-grade software used by both hobbyists and professional animators.
Dragonframe: A full-featured software system suitable for ambitious teens for making high-quality stop-motion animation.
iMotion: A free time-lapse and stop-motion app for iOS
iStopMotion Remote: A free app for iOS
Lapse It: Free time-lapse video software for iOS or Android
Stop Motion Studio: An inexpensive app designed for making stop motion videos on your Apple iPhone, iPad, or desktop Mac.
iKitMovie: Free stop motion animation software for Windows.
Hue HD: An animation studio designed for the classroom.
OGOBILD Animate It: Ogobild is a line of creative construction toys popular in the UK. Their Animate It Studio Kit is an all-in-one kit for kids learning how to create their own stop motion animated movies. It includes the Animate It! Express software, which was developed with the Aardman Animations studio (the creators of Wallace & Gromit).
Stopmotion Explosion: This kit contains everything needed to create stop motion films. Users can either follow the instructions contained in the Stopmotion Explosion book, or watch exclusive video tutorials in the Stopmotion Explosion Content Hub.
Stopmotion Studio 4: This kit provides software, but requires you to supply your own camera. Many DSLR cameras can work for this purpose.
Van Aken Claytoon Clay: Special modeling clay that does not harden, and thus is well suited for using in stop motion animation.
As an additional resource, here is a review of various stop motion kits.
Advice and Instruction
Helpful articles on making stop motion movies:
The Klutz Book of Animation