What does it mean when Headlights of a 2000 neon flash on and off when car is turned off?

In most vehicles, including the dodge neon, lights flashing on and off generally means that there is a malfunctioning vehicle security system, or the car's alarm has been triggered. When the vehicles ignition key slot is in the off position and the key is removed and the vehicle has been secured by closing the door while the door's lock is in the lock position or with a keyless remote control, the alarm system becomes active. There are many different types of alarm systems and most that have a relay that controls the headlights are installed by the vehicles manufacturer. I would recommed asking your dealer how to disable the system via a "valet switch"... that should stop the lights from flashing. If that doesn't work you could remove the headlamp fuse with a pair of needlenose plyers but I wouldn't recommed this because your wouldn't be able to use your headlamps until you return the fuse and you may disable something else that you didn't intend to disable because it's on the same circuit.