Tell it to Sir Paul. This is from the official Beatles website:
Lennon’s son, Julian, inspired the song with a nursery school drawing he called “Lucy – in the sky with diamonds”. Shortly after the song’s release, speculation arose that the first letter of each of the title’s nouns intentionally spelled LSD. Although Lennon denied this, the BBC banned the song.
In a 2004 interview, Paul McCartney said that the song is about LSD, stating, “A song like ‘Got to Get You Into My Life,’ that’s directly about pot, although everyone missed it at the time.” “Day Tripper,” he says, “that’s one about acid. ‘Lucy in the Sky,’ that’s pretty obvious. There’s others that make subtle hints about drugs, but, you know, it’s easy to overestimate the influence of drugs on the Beatles’ music.”
Note that the first paragraph implicitly explains Lennon’s denial of the obvious at the time. It may well be that they got the title from wee Julian’s drawing, but there’s a reason they picked that particular drawing to immortalize in song.