Lennon had been inspired by the memory of his mother Julia singing the song Wishing Well from Disney's first full-length animated film Snow White and the Seven Dwarfs. This may seem to be an unlikely source for a brash young rock-and-roller, yet it only serves to demonstrate that Lennon and McCartney had a broad musical knowledge even at the start of their songwriting partnership.
It was not an automatic choice for George or the Beatles, however. Lennon famously recorded a demo of the song in late '62 for fellow artist Billy J. Kramer in a Hamburg bathroom stall and pulled the chain on the toilet once he had finished. Kramer did wind up recording the song with his backing group the Dakotas a month or so after the Beatles, and his version went to the top of the charts in the UK (number one on some charts and number two on others).
After the Beatles recorded the song, they included it as George's vocal spotlight in their stage act for the next few months of 1963. He also performed it on multiple TV and radio appearances made by the group during this period. By the summer, they felt the song had run its course and replaced it with George's faithful cover version of Chuck Berry's Roll Over Beethoven.
The American label Vee-Jay Records initially released Do You Want to Know a Secret on the album Introducing...the Beatles. Once Beatlemania reached the US, the label chose the song as the A-side of a single in March of 1964. American fans bought this single in such quantities that it went all the way to number two on the Billboard chart, an impressive feat for young Mr. Harrison.