There are a few theories. They first called themselves The Beatals which was a take on Buddy Holly and the Crickets. The Long John and the silver beetles was suggested and silver beetles settled on and eventually became the Beatles. Another was from the film The Wild Ones there was a motorbike gang called the Beetles and they named themselves the Beatles because they were a beat group.
John Lennon was a big fan of American rock & roll, as well as word play (he loved Jabberwocky). While trying to come up with a name for his band, he reflected on Buddy Holly and the Crickets, and noted that that would've been a great name for a British band, as it would have a double meaning. He started considering other "insect" names, and settled on beetles. He decided to spell it Beatles, to symbolize "beat" music (as rock was often called in those days).