The "Voldemort will mark him as his equal" part was a little trickier, but then again it wasn't clear how it related to Harry, either. Neville had a messed up relationship with his parents, mainly due to the fact that Voldemort had them tortured into insanity. Voldemort had some serious mommy issues, too, so in that sense Voldemort made him his equal.
But the best part of the theory, the part that makes it all come together, is the idea that Dumbledore, the principal of Wizard High, was intentionally allowing Voldemort to think Harry was the chosen one just to protect Neville. While we're not saying they should have just ditched Harry in the last book and turned the other kid into the protagonist, it would have been pretty cool to have Harry find out he wasn't the center of the universe after all, if only because it would have made the character slightly less insufferable.
"Now that I know I'm not the chosen one, I can finally get my dick tattooed."
The later books do flirt with the idea, however briefly. And it would have fit right in with Rowling's habit of pillaging the first three books to turn all the one-off magic items and half-serious gags into game-changing MacGuffins.