After meeting John Lennon, what song did George Harrison play to demonstrate his ability?

Lennon asked Harrison to play "Raunchy." Only after hearing him play that did Lennon accept Harrison into their group.

There are many stories about the formation of The Beatles, e.g. The Beatles Anthology (and a few other books stored outside in the garage) make no reference to any "audition" or "demonstration" by George Harrison. Another example:

Although George met the group early in the year he was not immediately made a member of the group. In fact, he was never "officially" asked to join but became a member through perseverance. He would sit in at rehearsals and perform with the group when one or another of the group's guitarists didn't show up. He eventually replaced Eric Griffiths when it was decided that the group had too many guitarists. Colin Hanton, who was a good friend of Eric's, was asked by Nigel Whalley to go along with the group and Eric was "frozen out" of the Quarry Men, by the simple method of not informing him of the time and location of rehearsals.

http://www.beatlesexaminer.com/namec.htm

Nonetheless,this may be the answer (whether it is a verificable fact is another question): George Harrison playing Raunchy by Bill Justis. Source: The Rough Guide to The Beatles by Chris Ingham

http://books.google.com.au/books?id=htl2U1fPq8QC&pg=PA6&dq=harrison+1958+raunchy&sig=cmD5e93c4G4G7nUDkMVAoRQ9jb0

This Wiki article agrees with "Raunchy" as being the song that earned Harrison a place in the band.

Its references for this assertion are:
Chris Ingham (2 November 2009). The Rough Guide to the Beatles. Rough Guides Limited. p. 12. ISBN 978-1-84836-752-4.

Philip Norman (17 May 2011). Shout!: The Beatles in Their Generation. Simon and Schuster. p. 91. ISBN 978-0-7432-5378-9.
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In 1958 a then fourteen-year-old George Harrison performed the song to John Lennon and Paul McCartney on the top deck of a bus, and was so note-perfect Lennon decided to let him into his band, the Quarrymen, which later became the Beatles, despite earlier reservations about Harrison's age.

https://en.wikipedia.org/wiki/Raunchy_(instrumental)