The Adoption Theory
Not much to say in regards to this theory. Basically, the theory derives from Rita's comment in "Ties That Bind." In the episode, Lincoln eavesdrops on his parents' conversation, concluding that they intend on throwing them out, when in actuality, they were actually arguing on his father's tie collection. When Lincoln overhears them one last time, he hears his father say that he may "throw out the white one" as he didn't think it was his. Lincoln interprets this to mean that he was the one selected to be thrown out, unaware that it was referring to a rabbit tie.
Towards the end of the episode, while it's made clear that none of them are being thrown out, Rita states that she and her husband loved all ten of them, and her husband corrects her. While many took this to as a joke, some went a step further, and speculated that Lincoln was the only non-biological child of the Loud family. This would explain many things, namely the color of his hair.
However, the theory was debunked in Not a Loud when Lincoln's birth story was revealed.