OK so it is the connector going into the Arduino's Gnd pin between pin 13 and Aref that melted?
Their is no way that should have melted because it does not carry much current at all.
So we are left with the conclusion that either:-
Heat generated from another source melted it.
Or
There was some sort of error in the wiring that forced a lot of current through that pin.
But missing from that diagram and your photographs is how you powered the Arduino, it looks like that is the clue to this excess current.