The most famous band name in pop music is a bug joke with a spelling mistake built in on purpose. It also took the group about four years and several other names to get there.
The band began in November 1956 as the Quarrymen, a skiffle group John Lennon formed with friends from Quarry Bank High School. The name came from a line in the school song: "Quarry men old before our birth." Paul McCartney joined in 1957 and George Harrison in early 1958.
The names kept changing. By January 1959 the three guitarists were playing as Johnny and the Moondogs, and for a while as the Rainbows.
The insect idea arrived in January 1960 with Stuart Sutcliffe, an art-school friend of Lennon's who bought a bass. According to Wikipedia, Sutcliffe suggested "Beatals" as a tribute to Buddy Holly and the Crickets. Etymonline gives the same model: beetles, after the Crickets, with a pun on the musical sense of "beat."
Who deserves the credit is less certain. Etymonline says the name was "supposedly" Sutcliffe's. Other accounts say Lennon and Sutcliffe worked it out together, and Lennon did little to settle it.
The group tried several versions in 1960. They were Beatals until May, then the Silver Beetles for a tour of Scotland. By early July they were the Silver Beatles, and by the middle of August simply the Beatles.
Lennon enjoyed muddying the story. In 1961 he wrote a comic history of the band for the Liverpool paper Mersey Beat. In it he claimed the name "came in a vision." A man appeared on a flaming pie, he wrote, and told them: "From this day on you are Beatles with an A." McCartney later borrowed the joke for the title of his album Flaming Pie.
Other theories circulate. One points to the 1953 biker film The Wild One, in which a rival gang is called the Beetles. The Beatles Anthology mentions this story. But the film was banned in Britain until 1967.