FIG. 1 is a top, right side perspective view of a new, original and ornamental design for a hair styling apparatus handle;
FIG. 2 is a top plan view of FIG. 1;
FIG. 3 is a bottom plan view of FIG. 1;
FIG. 4 is a left side elevation view of FIG. 1;
FIG. 5 is a right side elevation view of FIG. 1;
FIG. 6 is a rear elevation view of FIG. 1; and,
FIG. 7 is a front elevation of FIG. 1.
The broken lines in the drawings depict portions of the hair styling apparatus handle that form no part of the claimed design. The broken lines in the drawings further depict a styling portion of a beauty tool and a power cord connected to the handle as environmental subject matter that form no part of the claimed design.