Old myths take a long time to die. There was a lot of controversy around the meaning of the title for awhile and apparently still exists. It was speculated that the title’s initials, LSD, was a reference to the drug. John Lennon denied it. His son Julian defended his father saying that he did indeed paint a picture of classmate Lucy and titled it that. The actual Lucy told the story of the day Julian painted the picture while sharing an easel at school.
But that is just the title. The lyrics may be a whole other matter. John Lennon claimed that “It is not an acid song. The imagery was Alice in a boat.” I’m no expert, but I assume this is a reference to the boat ride that Alice Pleasance Liddle took with Charles Lutwidge Dodgson (a.k.a. Lewis Carroll) which inspired him to write Alice’s Adventures in Wonderland and Through The Looking Glass and What Alice Found There.
When Paul McCartney, who also received credit for contributing to the creation of the song, was asked if it pertained to drugs, the response was, “It’s pretty obvious.” In my opinion, his answer is ambiguous. While it implies that the lyrics were drug-inspired, this has never been publicly stated by by either Lennon or McCartney as far as I know.