Poor, poor Jessie. I don't think she'd be very happy to learn that Cassidy got so many more theories than she did.
So here's... err... something.
Jessie's hair is actually a Pokemon. This Pokemon, an extremely rare species called Harrier, clings to her head and feeds on her real hair. Jessie herself doesn't realize this, since it's been around as long as she can remember. That's why her hair grows back so quickly when cut: Harrier gets detached from Jessie's head, then wanders back and reattaches itself when nobody is looking. It sags a bit when it gets soaked through, but that implausible curve is actually its natural shape.
The whole thing started years ago, when the Team Rocket scientists were trying to develop super Pokemon-human hybrids. The first step in the process was to genetically engineer a new Pokemon that would attach to humans and develop a symbiotic relationship with them. At first Project Poke-Symbiosis seemed to be a dismal failure. The human test subjects were grunts who had been given lab duties as a punishment for wasting money on missions. None of them were able to properly bond with their Harrier; they just had too much trouble adjusting to having enormous objects stuck to their heads, and all of the threats of beatings and imprisonment in the world didn't seem to do a thing to help. The Harrier, feeling terribly unloved, would inevitably get depressed and detach themselves. But one clever scientist speculated that if they attached the Pokemon to a baby, that baby would grow up loving the Harrier as part of his-or-herself, never knowing how much easier life is when you don't have a massive weight clinging to your head. Since Team Rocket specialized in stealing Pokemon rather than humans, babies were in somewhat short supply. But as luck would have it, it just so happened that one of the elite agents had recently popped out a kid. So one day while Miyamoto was out counting her money, a team of grunts snuck into her apartment and attached a Harrier to her infant daughter. Miyamoto was somewhat surprised to see that Jessie's hair had grown by over a foot over the course of an hour, but she never did bother to give it much thought. Jessie grew up adoring her hair, never knowing that it was actually a Pokemon, and Harrier flourished under her loving vanity. Sadly, Jessie was quickly lost to the foster care system, so the scientists never did learn that their symbiosis experiment was a success.
Harrier talks to Meowth sometimes, but it's threatened to strangle him in his sleep if he tells a soul about it existence. So far, he's managed to keep the secret.
Now come on, surely somebody can come up with a better theory than that.
