My absolute favorite fan theory is that Hermione is actually Harry's older sister. That she was placed, for her protection, with the Grangers, and it is because of her that Harry is safe at Hogwarts (the protection charm, like the one placed on Privet Drive), and that she knows it.
This site does a great job of breaking down the theory, but I'll share some of the highlights.
*They are 10.5 months apart in age (Hermione was born the September prior to Harry).
*Hermione's overall protective nature towards Harry, even going so far as to answer for him.
*The fact that the Granger's are not explored as characters, like the Weasleys, Neville's grandmother or even Hagrid's family.
*How seemingly easy it was for Hermione to drop everything and help Harry, to the point of ruining a vacation with her "parents".
*How she knew almost everything about Harry when they first met, even though she'd only known she was magical for about 2 weeks.
*No romantic feelings on either end towards each other
Now, I don't know about you, but this made my heart happy and brain explode just a little. I will now wait patiently for JK Rowling to announce on Pottermore that this is true and make my world a little brighter.