Broken Gadget is a “gadget” that does nothing whatsoever. It is used as a placeholder for when:
Per standard form, it is also given by gadgets with “(once)” in the description, which manually change themselves to Broken Gadget upon use.
It can also be assigned to equipment as an actual gadget, though in-game this is indistinguishable from the equipment having no assigned gadget. This is a bizarre choice that only occurs in modded equipment (such as “Rubber Duck” from Halloween Special, or “Scrap Remote” from More Fluff).