Today is Mother's Day in Portugal and my mother is the most important person in the world to me so I'm doing a special Mother's day to celebrate her! This chewing gum wrappers always left me thinking so I went to look for answers.
Neville was raised in the care of his grandmother, Augusta Longbottom, Frank's mother. Neville regularly visits his parents for Christmas and other holidays. In St. Mungo’s, though his parents do not overtly recognise Neville on these visits, Alice Longbottom, Neville’s mum always gives Neville some chewing gum wrappers. His grandmother tells him to throw it away, but he pockets it. Do the gum wrappers mean anything? Of course they mean a lot.
They show how Alice’s brain is broken beyond any healing. She can’t realize that she’s giving her son something of no value.
They show that, no matter her brain condition, Alice’s love for Neville is still there, strong and unending. She may not know the value of the gum wrappers, but she feels the need to offer something to her son—an offering of pure love. Love is more powerful than madness.
They show that Neville loves his mother dearly, since he’s willing to keep dozens of gum wrappers in his room, just to remember her and feel her love. He wouldn’t want to show (even to himself) that he threw away something Alice had given him, no matter what it is.
JK Rowling answers this in an interview:
Q:What is the significance, if any of the gum wrappers that Mrs. Longbottom keeps giving Neville? JKR: That was also asked of me this morning. That idea was one of the very few that was inspired by a real event. I was told what, to me, was a very sad story by someone I know about their elderly mother who had Alzheimer's, and the elderly mother was in a closed ward. She was very severely demented and no longer recognized her son, but he went faithfully to visit her twice a week, and he used to take her sweets. That was their point of connection; she had a sweet tooth, she recognized him as the sweet-giver. That was very poignant to me. So I embroidered the story. Neville gives his mother what she wants, and (it makes me sad to think of it) she wants to give something back to him, but what she gives back to him is essentially worthless. But he still takes it as worth something because she's trying to give, so it does mean something, in emotional terms. But, the theories on the sweet wrappers, are really out there. Q: You can’t blame them. JKR: I mean she's not trying to pass him secret messages. Q: She's not really sane
JKR: No. You're right. But that's a classic example of, "Let's just shut that one down," because it doesn't really lead anywhere very interesting even if they're wrong. Q: It's probably one of the most touching moments in the books. JKR: I think it is important as a character moment. Font: http://www.accio-quote.org/articles/2005/0705-tlc_mugglenet-anelli-3.htm