Change WordPress user roles and capabilities › Forums › Restrict or Permit access inside WordPress – how to › How to allow backend users to edit the menu (but not themes etc.) › Reply To: How to allow backend users to edit the menu (but not themes etc.)
You may always get information about capabilities in use opening “Admin menu” window for the “Administrator” role. Menu items under “Appearance” menu are protected with ‘edit_theme_options’.
Add ‘edit_theme_options’ to your role, update it. Then open “Admin menu” for this role and block unneeded menu items for it.