"Think of the children" (also "What about the children?") is a phrase that became a strategy used when you are in an argument with someone. Basically in its simplest use it can mean just children's rights, like for example talking about child labor problems. But when this phrase is used in the middle of an argument, it can be used to try to make people feel emotional when thinking about the needs of children, and then to try and fool people into doing what they want because they might be scared that they do not want to harm children.