For some reason, when I need a leading child character in an Oz story, I tend to choose Button-Bright.
When I downloaded a program to create simple text-based adventure games, the first game I made with it was called "My Adventure in Oz," in which the player was Button-Bright, gathering guests for Ozma's birthday party. (The play scheme would be that the character had lost something, so you had to find it and give it to them, then they'd join you.) As it's currently available for download (DOS game, so Windows only), I guess it's really the first Oz story I published.
Later, I tried my hand at writing an Oz story that found Button-Bright and the Shaggy Man traveling through Oz. The story was going to involve a magic fountain that was in fact, the Fountain of Youth, which would be revealed as the reason why people stayed young in Oz. (The idea being that's all it would do, being diluted in normal bathing and drinking water.) The fountain would be guarded by the Lonesome Zoop from the Oz Film Manufacturing Company's The Patchwork Girl of Oz silent movie, introducing the character into the more mainstream Oz universe. There was also going to be a magic ruby ring that blocked enchantments, and some rarely-seen Oz characters, including my original character Perry, the son of ex-General Jinjur. (Since he was born in the Munchkin Country, "Perry" was taken from periwinkle flowers.)
That story was abandoned when it got bogged down and I couldn't decide what to do next.
Later, I started a pass-around story at the IWOC Forums. The idea was, the story would be added to by forum users. Rather shamelessly, I included many elements from my abandoned story, starting with Button-Bright and the Shaggy Man, and later adding in Perry and the Ruby Ring, which, instead of blocking enchantments, absorbed magic around it. The Ruby Ring became such a key plot device, I proposed (and the few people who were still adding agreed) that we call the story "The Ruby Ring of Oz." We actually finished it, and the big, 61-page thread can still be viewed at the IWOC Forums. (One of the contributors and I have discussed that it needs heavy editing before anyone should attempt to publish it, with some characters biting the dust.)
But Button-Bright... He is a fun little chap. He's also, I believe, the only character Baum introduced in an Oz book who became a main character in a non-Oz book, and one of the few who definitely matured from his first appearance to his next one.
All readers of the Oz books know Button-Bright first appeared in The Road to Oz as a young boy, who was described to be about two or three years younger than Dorothy. John R. Neill's depiction of him may vary from five to seven, perhaps, but not any older than that, I'd say. As Dorothy's age seems to be a matter of debate among Oz fans (according to one book I'd read, Button-Bright would have been a teen or pre-teen about the time of Road), Button-Bright's age is undetermined. I like the idea that Dorothy was about ten or eleven by this time, so Button-Bright would probably be seven.
Button-Bright was characterized in this book by the sailor clothes he wore, and his trademark phrase, "Don't know." Really, the only information we knew about him was that he lived with his father and mother, and that their back yard had a well in it, suggesting they lived in or around a town or city. Somehow, he managed to appear outside of Foxville in time to meet Dorothy, Toto, and the Shaggy Man. One other major factor in the story is that, because King Renard (or Dox) of Foxville gives him a fox's head, he had to restore it in the Truth Pond, which also made him tell the truth. In Baum's books, Button-Bright had always been honest, so the Truth Pond had little effect on his personality. (In one of Eric Shanower's stories, he explains that Button-Bright's youth was also a safeguard against offending someone with the truth.) Later, the Wizard sends him home in a giant soap bubble.
Button-Bright later appears in Sky Island, the second book containing the adventures of Trot and Cap'n Bill. Here, Button-Bright is obviously older than before, about the age of Trot. We discover where he lives, Philadelphia, having now lost the trademark "Don't know," phrase. He does still ask questions, though. His family appears to be well-off, as Trot notices his fine clothes. Perhaps his family came into some money or an inheritance, or his father was able to better provide for his family. He even reveals his real name, Saladin Paracelsus de Lambertine Evagne von Smith.
Button-Bright's apparent aging causes us to ask how long it has been since the events of The Road to Oz. If Button-Bright was seven in that book, he might be ten or eleven by this time, making it three or four years, plenty of time for big changes in Button-Bright's family and himself.
Button-Bright is more of a major character in Sky Island, as it is his magic umbrella that carries Trot, Cap'n Bill, and himself to the Island. The umbrella can carry it's carrier to wherever they wish by flying, and later turns itself into a charging elephant.
So, L. Frank Baum beat P. L. Travers with flying umbrellas. Button-Bright beat Mary Poppins to it. (In the book Visitors from Oz by Martin Gardner, he claims that Mary Poppins now owns Button-Bright's umbrella, and that it's head changed into a parrot for her. However, this is the same book I mentioned above where, by it's aging of Dorothy, Button-Bright would have been a teen or pre-teen by the time of Road.)
Button-Bright returns to Oz in The Scarecrow of Oz, when Trot and Cap'n Bill find him in the Land of Mo, where he has lost his umbrella. This book added the final touch to Button-Bright's character: he was always getting lost. He is not such a key character in this story, though he does lead the Ork back to Trot, causing the Ork to help in the climax of the drama in Jinxland. He stays in Oz with Cap'n Bill and Trot, seeming to be content not to return home.
This always struck me as odd. Wouldn't his parents worry that their son had gone missing, and then he never turned up again? And Button-Bright doesn't care? Or suppose something very serious had happened and Button-Bright decided to escape with the umbrella. Maybe his parents died? A bit far-fetched, I must admit, but it might explain his complete reluctance to return home.
While he returns in The Lost Princess of Oz, joining Dorothy and the Wizard's search party for Ozma, he continues his habit of getting lost constantly, which ends up being key to resolving the plot, since it is, in the end, he who finds Ozma. (He just didn't know it.)
It had been established in The Lost Princess of Oz that Button-Bright became friends with Ojo, Baum's fourth leading boy character in Oz (after Tip, Zeb, and Button-Bright himself), and a brief mention in The Tin Woodman of Oz says that the two play together.
Button-Bright is also part of the rescue party for Dorothy and Ozma in Glinda of Oz, but he doesn't have any real bearing on the plot, nor do we discover much more about his character.
I recently commented on the IWOC Forums that Button-Bright has a short attention span, perhaps ADHD. No one else has agreed that he might, but Sam Milazzo commented he might have Asperger's syndrome. ("Strasheela" said it's possible, but he didn't specify if he meant my idea or Sam's.)
But really, in the end, it doesn't seem to matter, as Button-Bright is quite an endearing character, and while he might not rank highly on lists of favorite Oz characters, he certainly is a good one.