The right side of the brain, I think has two different meanings in the song. At the beginning he is talking about the girl who never used the right side of her brain as in the actual right side of her brain which deals with emotion and romanticism. However when he starts talking about himself in the, "the right side of my brain", I think he is actually referring to the right side as in the "correct" side, the left side which deals with logic, meaning he wasn't thinking logically about the relationship when he should have. In short, the girl was being too unemotional in the relationship, using the left side of her brain, while the guy was foolishly thinking romantically, using the right side of his brain.
The right side of the brain, I think has two different meanings in the song. At the beginning he is talking about the girl who never used the right side of her brain as in the actual right side of her brain which deals with emotion and romanticism. However when he starts talking about himself in the, "the right side of my brain", I think he is actually referring to the right side as in the "correct" side, the left side which deals with logic, meaning he wasn't thinking logically about the relationship when he should have. In short, the girl was being too unemotional in the relationship, using the left side of her brain, while the guy was foolishly thinking romantically, using the right side of his brain.