I claim the ornamental design for a backrest stanchion for a child's chair, as shown and described.
FIG. 1 is a front perspective view illustrating a backrest stanchion for a child's chair;
FIG. 2 is a rear perspective view thereof;
FIG. 3 is a front view thereof;
FIG. 4 is a rear view thereof;
FIG. 5 is a right side view thereof;
FIG. 6 is a left side view thereof;
FIG. 7 is a top view thereof with broken lines disclosing elements not part of the claimed design;
FIG. 8 is a bottom view thereof;
FIG. 9 is a front perspective view thereof with broken lines disclosing the environment associated with the design but not part of the claimed design; and,
FIG. 10 is a rear perspective view thereof with broken lines disclosing the environment associated with the design but not part of the claimed design.