I do not think it was arbitrarily chosen. I think there is a universal aspect to why blue was chosen to mean someone was sad or depressed.
http://www.colour-affects.co.uk/psychological-properties-of-colours
BLUE. Intellectual.
Positive: Intelligence, communication, trust, efficiency, serenity, duty, logic, coolness, reflection, calm.
Negative: Coldness, aloofness, lack of emotion, unfriendliness.
Blue is the colour of the mind and is essentially soothing; it affects us mentally, rather than the physical reaction we have to red. Strong blues will stimulate clear thought and lighter, soft blues will calm the mind and aid concentration. Consequently it is serene and mentally calming. It is the colour of clear communication. Blue objects do not appear to be as close to us as red ones. Time and again in research, blue is the world's favourite colour. However, it can be perceived as cold, unemotional and unfriendly.