Poor connection, dirty contacts...creates resistance, which creates heat. No resistance, no heat generated, no burns. If that pin was a bit recessed, maybe it didn't have as much contact area as the rest of the pins and all that current was forced to go thru a smaller contact area than the rest of the pins.
If the monitor was actually drawing 12v @ 5amps as you say, that's 60 watts. Even a little bit of resistance can make heat over time.
In short, I would venture to guess that it wasn't an issue with the connector, or the pins, or the wiring, but more of a mechanical issue with the pin contact itself.