It is commonly believed that where there is a problem or disagreement between people, even if they are of good will, the "solution" must be either one person wins and the other(s) lose, or vise versa, or that there must be a compromise, where each person gives up something and everyone ends up losing.
Taking Children Seriously denies this idea. According to TCS, it is possible to solve problems and disagreements between people of good will, in such a way that everybody wins, and no one "gives up" anything.
A common preference is a solution which everybody involved prefers, not just to the initial ideas others involved had come up with, but to all other candidate solutions they can think of, including their own initial idea.
A common preference is a preference that is common to all parties. It is (usually) a new idea that had not been thought of at the outset of the discussion. It is a solution that each and every party genuinely, wholeheartedly prefers.