link to addon: http://www.blackcrow.eu/blender/addons/object_to_mesh_name.py
put the file in …/Blender Foundation\Blender\2.79\scripts\addons
enable the addon in user prefs under object “Object to mesh name”
select the objects you want to rename (or press A to select everything), press spacebar and look for
press enter and all your mesh names are now the same as the object it’s parented to.