Snow White, because someone mentioned it, would be "Don't let fear rule you." The biggest lesson she learned was not to be fearful and let that rule her to run away. As soon as she conquered her fear, things became better.
My daughter loves Toy Story for the friendship between Buzz and Woody, she won't ever put one without the other.
The Incredibles I would say "Don't Hide Your true self.'
Meet the Robinsons is "Always move forward" and don't be afraid to fail, exalt in failure (this was also in Chitty Chitty Bang Bang.)
Lion King is about responsibility and family. Same with Mary Poppins too.
The Three Lives of Thomasina would be about family, letting go of mourning, and opening yourself to new happiness.
So Dear to my Heart is about hardwork and perseverence even when it doesn't seem like you'll ever win.
The Sword in the Stone is about education being the true power and strength, the strength of kings.
The Black Cauldron is about friendship, loyalty and responsibility.
Tarzan is about family and melding worlds.
The Great Mouse Detective is about using your brains to figure out the hardest problem, no matter how little and insignificant you think you are. Oh, and positive training the dog because let's face it, no mouse could train a dog using a choke chain. :P
Old Yeller is about loyalty and giving your life for your family, as well as responsibility and doing what must be done even when you hate it.
Dumbo is about finding your own special talent to succeed in life.
Pinocchio is about being good and not shaming your family.
Finding Nemo is about not being afraid of a little adventure, and don't count your parents out.
Robin Hood is about doing what's right even if it goes against the government rules.
The Cat From Outerspace is about friendship and loyalty.
The Princess and the Frog is about working hard for what matters, and recognizing what really matters...