The ornamental design for a head massage device handle, as shown and described.
FIG. 1 is a side view of a head massage device handle showing my new design;
FIG. 2 is a right end view thereof through which flexible massage fingers that are not shown extend; and,
FIG. 3 is an exploded view thereof.
The broken line showing of the flexible fingers in FIG. 1 and the handle portion in FIG. 3 is for illustrative purposes only and forms no part of the claimed design.