If we ever figure this out I'll try to let you know. All the BH and Landmarks have this welcome light, just not magical like ours has become.
The Welcome Home Light is powered by an electronic timed relay (also known as a pulse timer in some cases). The timer activates the relay which turns on and off your light. The door switch is actually completing a ground circuit which activates the relay turning the light on. There is a diode in line somewhere between your light and the manual switch to keep power from going back to the manual switch when the light is activated by the relay.
Because the light is coming on and off by itself it could only be one of two things. The door switch as Michael suggested is wet OR the timer is finding a ground prior to the door switch. Either one will activate the timer and it will work as designed-turning the light on until the timer is "timed out" and turns the light off. Once it goes out, it starts all over again.
The second thing is the timer relay could be bad but chances are if that was the case the light just wouldn't work.
The fact that this started after pulling in the rain may indicate that water has intruded past the contacts of the door switch and into the switch itself allowing the ground to be completed. A hair dryer blowing on the door contact switch may solve the problem.
There is an off chance that the door switch is not completing the ground but in fact acts as a break on the positive side and when activated allows the power to flow from through the switch and then to the timer. Still, if that is case, water will do the same thing. I have not been behind the basement wall on our trailer to know how HL has the light wired but have installed several pulse timers in trailers. Simple circuit but water can make low voltage electricity do some really strange things.
Hope this helps.