Two good children's monologues are the Cowardly Lion's first encounter with Dorothy in the movie "The Wizard of Oz" and Dory's plea to Marvin to stay with her in the movie "Finding Nemo." Because a monologue is "a (usually long) dramatic speech by a single actor," according to dictionary.com, any material that a kid can deliver as a one-sided speech is a good monologue for the child to learn.
"The Wizard of Oz" has long been a favorite source of monologues for children. Dorothy, Scarecrow, Tin Man, Cowardly Lion and the Wicked Witch provide kids with their choice of larger-than-life characters to become for a minute or two. For example, a kid who likes moving a lot while speaking is sure to enjoy feinting punches while delivering the Cowardly Lion's lines on meeting Dorothy: "Roooaaarrr! Put 'em up, put 'em up! Which one of ya first?"
A more recent source of monologues for kids that's become popular is the Pixar animated movie, "Finding Nemo." It too has a selection of vivid characters with a variety of personalities for children to step into. One especially challenging and moving monologue would be forgetful Dory's speech begging Marlin not to leave her: "...I look at you, and I...and I'm home! Please, I don't want that to go away."
Other than movies, some good monologue sources worth exploring are television programs and plays produced for children, bookstores, acting coaches, acting classes and workshops, and Internet websites that collect monologues. Whatever character type a child relates best to, there's a monologue perfect for that kid to learn. Don't be afraid to encourage kids to try writing their own monologues. This may be the ideal time for a child to tell a funny story from summer camp or a family holiday.