Long answer: Keep reading :)
Last month my editor for my forthcoming picture book, Rice from Heaven, said she wanted us to create a subtitle just to make it clear what it's about. So the editor and her team suggested some, my agent suggested some, and then I reached out to a couple of my critique groups, and they suggested some. So I had about 10 titles to work with.
A subtitle, IMO, should be written concisely, and beautifully sum up the main idea of your book.
Here are some examples of recently published picture books with subtitles:
Vincent Can't Sleep: Van Gogh Paints the Night Sky by Barb Rosenstock
Long May She Wave: The True Story of Caroline Pickersgill and Her Star-Spangled Creation by Kristen Fulton
She Persisted: 13 American Women Who Changed the World by Chelsea Clinton
After the Fall (How Humpty Dumpty Got Back Up Again) by Dan Santat
Kick! Jump! Chop!: The Adventures of the Ninjabread Man by Heather Ayris Burnell
La, La, La: A Story of Hope by Kate DiCamillo
Did you notice the first three were nonfiction picture books, and the last three were fiction picture books? So subtitles can be used for both.
Author Darcy Pattison has something interesting to say about subtitles here. Will the subtitle help teachers, parents, and librarians find your book in the database?
Lastly, this article has good points about the dos and don'ts of subtitles.
So--what is my subtitle? You'll have to wait for the cover reveal to find out!