In keeping with the "theme" of June, the book I'm recommending is Anne of Green Gables by LM Montgomery.
After I got back from my vacation to Prince Edward Island, I decided I wanted to reread Anne to see how being there has changed the way I feel. And I was reminded by how much I love Anne and the Cuthberts. There is something about that little red headed girl who worms her way into everyone's hearts, the reader's included. I think every little girl should be introduced to Anne Shirley, because of her imagination, and her desire to be good. She has faults (stubbornness), and makes mistakes (liniment in a cake, getting her friend drunk, breaking a slate over someone's head, dying her hair, etc.), which makes her even more endearing. And there are so many good things Anne has to say, some adorable, and some quite profound. Things that young girls need to hear.
That being said, I remember it being a lot more difficult to read than it actually is. I think I read Anne of Green Gables for the first time when I was about 12, and I did read it several times throughout the years, but reading it again it was a quicker read. This isn't a bad thing, just an observation and something to keep in mind if you are thinking about recommending it to a kid.
And to finish this review off, I cried at the end. Even though I knew what was coming, I still cried. I think that's the best recommendation I can give--that I can still have an emotional reaction to something after having read it (and seen the movies) several times.
The Alpine Path: The Story of My Career by LM Montgomery
This is a short autobiography (about 100 pages) where you learn about the inspiration of some things in Anne of Green Gables, as well as Montgomery's childhood, etc. Really interesting read.