Georgie renames Frederick the mule Long Ears, and other than Billy, Long Ears is her most constant companion throughout the book. Though she doesn't want him at first, believing she's paid for a horse and not a mule, Georgie eventually comes to value Long Ears, if only because they have been through some serious stuff together.
For all she dislikes his stubbornness, Georgie and Long Ears actually have a lot in common in that neither will budge from an idea once they get it in their heads. Georgie pursues the truth about Agatha as doggedly as Long Ears pursues Storm, the horse he's fallen head over hooves for. In the end, Long Ears becomes a symbol of valuing what's right in front of you. Georgie says:
After paying my IOU to the store, I bought Long Ears from the sheriff. I'd kept the Bechtler gold dollars because of my yearnings for my pa. But Pa wasn't coming back—like Grandfather Bolte—and I was tired of musing about situations that could never be. So I swapped daydreams for a true friend. The sheriff says he'll never call him Long Ears. But Long Ears knows his name, and so do I. (22.44)
Who needs daydreams when they've got Long Ears? Not Georgie, that's for sure. These two are two stubborn peas in a pod.