Many have wondered, "What was Dudley's worst memory? What did he see when encountered by those dementors in OotP?" All JKR has to say on the subject is this: "I think that when Dudley was attacked by the Dementors he saw himself, for the first time, as he really was. This was an extremely painful, but ultimately salutory lesson, and began the transformation in him."
Most people took this as JKR stating that Dudley finally saw himself as the total ass that he was, and that this led to the small amount of kindness Dudley displayed to Harry in Deathly Hallows. But what if this wasn't what she meant? Perhaps she was being cryptic (as usual).
I think Dudley saw a repressed memory of him doing magic and his mother and father reacting in a very negative manner. Maybe that's what Dumbledore was talking about when he sent that Howler to Petunia in OotP. It could easily have meant not only the obvious message- take care of Harry- but also another. I think Dumbledore and the Dursleys may have made a pact.
If they took care of Harry, then he would keep Dudley from getting his Hogwarts letter with Harry. The Dursleys definitely would have taken the deal!
I mean, what's the likelihood that Dudley would be a wizard? Looking at it from a scientific standpoint, the wizarding gene is apparently a dominant trait, as proven by the rarity of Squibs. Say that M is Muggle and m is Magic. If the two parents are both Mm (in other words, both magic, with a carrier Muggle gene), then there's a 1/4 chance for MM, a 1/2 chance for Mm, and a 1/4 chance for mm. That would completely explain the rarity of such a phenomenon- imagine this in a Pureblood situation. It would take the child of a Halfblood and a Pureblood to marry another Halfblood for this to even be plausible (and we all know how picky those damn Purebloods can be)!
However, I cannot apply this simple of a theory- it would require both Petunia and Vernon to be magic, which they obviously are not. But perhaps, scientifically speaking, the magic trait is similar to the eye color trait, with many different variations of the same trait. There isn't just one type of blue eyes, there are many. Perhaps magic works the same way; this would explain the difference in levels of power. Imagine a graph of the amount of magical power in Tom Riddle (powerful), Ron Weasley (average), Argus Filch (low amount, but able to be raised by courses like Kwikspell), John Granger (Muggle with an apparent magic gene), and John Doe (Muggle with no magic gene). It would likely have a steady negative slope, yes?
So, magic is as variable (and potentially changeable) as metabolism. This theory would work both for passing along magic, for creating Muggleborns seemingly 'out of the blue,' and creating Squibs (also seemingly from nowhere).
Back to Dudley...
I've established that there's a strong possibility he would have magic in him, the gene simply not strong enough to show through in either of his parents. So what's to stop him from accidentally showing it, and his parents having an incredibly negative reaction?
This would also potentially explain Petunia's multiplied negative feelings towards Lily. It's already established that she was jealous of Lily's ability; she also could blame Lily for making her child be a 'freak' she would be forced to envy. Granted, it was in no way Lily's fault, but it's always far easier to blame someone you hate anyway than someone you love (like we assume Petunia did her parents).
I'm rambling much further than my original intention: to state my theory on what Dudley saw that night. If any of you have concurrences or rebuttals (or just general statements- or your own theories!), feel free to comment with them.