I must confess that I'd never even heard this name before Kate asked about it. But, as a public service (for those who prefer not to wander over to Wikipedia--and don't bother now, since it's all here) here's what I came up with!
From Wikipedia, the free encyclopedia
Billy Shears was Ringo Starr's alias on the Beatles 1967 album Sgt. Pepper's Lonely Hearts Club Band, often credited as a concept album. Billy Shears is only mentioned in the title song and, implicitly, as the singer of the segued-into "With a Little Help from My Friends," and even more implicitly, in the title song reprise.
It is sometimes rumored that Billy Shears is in fact William Shears Campbell, supposedly a replacement for Paul McCartney after his "death."
Billy Shears was later mentioned in Ringo's 1973 hit "I'm the Greatest", written by John Lennon: "Yes, my name is Billy Shears / You know it has been for so many years."
In the 1978 RSO movie Sgt. Pepper's Lonely Hearts Club Band, a character called Billy Shears is played by Peter Frampton.
In the 1968 animated movie Yellow Submarine, John asks Jeremy "Who the Billy Shears are you?"