Carbuff - nothing worse than the day before Christmas eve to have a problem like that. What a pain.
I'm guessing. It sounds more like the 12V battery was low. Even though the lights come on, the radio plays, and all that good stuff - those things will work even if the battery is on the low side. It isn't like the old days with those power hungry tungsten incandescent headlights that took amps of power. All that stuff uses a lot less amperage now.
The key would have been the relays clicking. If when she tried to start the car, if there was either no click or lots of clicking - that would be the main relay not closing. That relay is in the under hood fuse box. I had a low 12V battery that did all the things you described - seemed like the car should run. But in fact there wasn't enough power to pull the main relay closed and it would just click like crazy.
Others have said it before - one of those lithium ion jumper batteries and a USB adapter for the 12V power outlet with a Voltage readout will get you out of a lot of trouble.
The car probably started on its own because the 12V battery got up to a critical voltage that worked. The rain and the extension cord had nothing to do with charging. The rain did have something to do with the 12V battery getting low - windshield wipers, maybe defroster and heater just hit it a little extra hard.
(If you're still on your original 12V battery and this is the 2013 in your signature, probably time to replace the battery).
I'm just guessing though.