Paul continues his argument that since we were set free we should not return to bondage of any sort: not bondage to "gods who are not God;" not bondage to the legalistic requirements of the religion of the Old Testament. We are not children of Hagar but Children of Sarah.
In the first part of the chapter, verse 4, Paul writes that we "might receive adoption as children." Most of us might read this "adoption" as lesser standing than the natural children of the family -- that is someone who was added on later. As I understand it (correct me if you know better), under Hebrew law the child by adoption had a greater standing than a child by natural process. The rationale being that the naturally born child was an act of God, and therefore not chosen. However, a child by adoption was a conscious and active choice of the head of the family. Having chosen, one may not change one's mind. A natural born child could, under the law be disowned. A child by adoption may not, under any circumstances, be disowned. In that way the child by adoption had a stronger standing.
I like being an adopted child of God. It means I wasn't random. I was chosen!