In the comic Mystique, with baby Kurt, was being chased by an angry mob. She threw her baby off a cliff (assumedly to kill him???). The baby was saved by his father, Azazel.
I don't think this storyline would work for this X-Universe. In this one Mystique is proud of being a mutant and she wouldn't run from an angry mob of non-mutants. She also wouldn't be ashamed of her son's appearance.
With that being said there are many different stories that could be used. For example, off the top of my head, we could see a flashback where we see Erik and Raven together, as we all expected after First Class. We could also see Azazel forcing himself on Mystique, or maybe she was tricked by him, or plain ole' cheats on Erik with Azazel.
Either way, 9 months later (or earlier, if it is seen during an ultrasound), it is proven that it is not Erik's child, but Azazel's. Either Mystique leaves to hide this from Erik, and/or from fear of Azazel taking the baby, or Mystique and Erik fight about this turn of events causing her to be rejected by him. Personally, I like the idea of her leaving to hide this from Erik, after finding this out during an exam (who would she have the exam done by?).
This could cause reasons for Mystique abandoning her baby. Maybe she does throw him from the cliff (since this child was the cause of her problems with Erik, or for a similar reason), though Azazel can't be the one to save him since he'd be dead by the time Kurt is born. Or maybe Mystique is the one to give the baby to Margali since she knows that she isn't ready to raise the child, and/or it will be too dangerous with what she is doing. Maybe she wants to go back to Erik and knows that knowledge of Kurt (being the product of Mystique and Azazel) would cause major issues, both for her personally, and maybe even for the Brotherhood.
What a great flashback this could be!!! Imagine we see some of this stuff above, and we see Mystique leave. Next we see her as Baroness Wagner giving birth, which would be a pretty interesting scene. And then we see her giving up the baby to Margali (at the circus). Flash forward almost 20 years and we see Kurt as the high flying act in the circus and a scene plays out that exposes him and which causes Professor X (and Beast?), who are there searching for this Mutant signature, to intervene, which brings Kurt to the school.
Wow, that would be AMAZING!!!!!
