Well, on the pages of Shazam! #4, the mystery is solved. In the story, "The Mirrors That Predicted The Future" we learn from the wizard Shazam that Billy's birthday is April 1st - no, I'm not fooling! The story begins by telling us, "It is Billy Batson's birthday, but as he walks down the street on what should be the happiest day of the year..." and we see Billy thinking to himself, "For twenty years time stood still for me...and I didn't celebrate any birthdays! Now, nobody even remembers what day this is!"
As the story continues we soon come to Shazam who is writing the following, "If the boy Billy Batson, who was born today, remains pure of heart...one day I will have him take over my life's mission...fighting evil in the world!" A look at the calendar on the wall indicates the date is April 1st.