The stone he put in its place looked almost exactly like Jon's stone, except that it wasn't. It was the Family Stone.
It worked just like the normal stone, except that wishes about an individual would affect the individual's immediate family. Biological relatives would be affected, but step-family, and adopted children would not.
For example, if Jon's wish was about himself, then it would also affect his mother, his father, his sister Zoe, his brother Mikey, as well as his grandparents, his aunts and uncles and his cousins.