Going back to the original comic series...
What's interesting is that the ooze that falls on the turtles was actually alien in nature and created by the Utroms (the alien race that Krang's design was based off from the 87 cartoon). So it's funny how people aren't really aware of the
FULL origin story and call foul when something seems a little off though there is a lot to the origin.
So to recap:
What happened: The 4 household pet turtles are involved in a freak accident which makes them not only fall from their owners hands but also get covered in a mysterious ooze when they end up in the sewer.
More backstory: The ooze was actually created by the Utrom, an alien race that look like brains with faces who control robots, who have been on earth since the 60's.
Food for thought: During the fight with The Shredder in the first issue of TMNT, Shredder beckons Leonardo and calls him "Kappa". Kappa is the word for river-child in Japanese, and here:
http://en.wikipedia.org/wiki/Kappa_(folklore).)
So all in all, the well-known origin story that everyone knows happened (the turtles getting oozed after falling into the sewer), the alien thing has some merit due to the origins of the ooze, and the Japanese folklore story also plays a part since Shredder is Japanese and would have been familiar with the folk story about the Kappa.