Wire connector

That is the correct connector. Your original one is brass, so just surface corrosion (green) you can easily remove. Get some spray "contact cleaner" (Ace?) and a set of small files (Harbor Freight). You can clean it in the housing. Then measure resistance from the connector to source (lamp switch terminal?). If <2 ohm, it should work fine. Re-install w/ dielectric grease (silicone, any auto parts).