You need to show a picture. I think (without seeing it) that you have but 2 choices of repair types. Remove the fabric with the bad seams and replace, or mend the seams from the top. If the pattern is such that you can applique something on top of the seams - even strips of seam binding tape or narrow sashing- in a color that will enhance your color scheme I would probably select that repair from the top method and sew through the whole quilt. It would in effect quilt the quilt while stitching the new fabric down. Since it is a child's quilt, might it be possible to select applique
shapes appropriate to childhood to cover the seams? Maybe a bad suggestion since I can't see the number or size of the frayed seams.