The cold weather severely hampers a battery's ability to give amperage. First cold night in winter time is when a lot of people find out that their weak battery (that's been working fine through the nice warm summer months) is actually dying. You can have as much as a 30% difference between cold cranking and warm cranking amps that a battery can deliver.
I'd say to get the battery tested, find out if it's weak. If it is, obviously replace it. If it isn't, make sure your starter solenoid is switching properly.
Directions on how to test your starter solenoid:
http://www.cygnusx1.net/Supra/Library/TSRM/MK3/manual.aspx?Section=ST&P=12
If that's working properly, and you still don't hear the clicking, you're going to have to check ignition switch connections.
On another note, does anyone really know why there is a starter relay with how they have it wired up? I looked at the wiring diagram and it looks like the same wire that's running from the ignition to the switch side of the relay is also the wire for the power side of the relay. So when you're turning your key you're using the same wire to supply power directly to the starter solenoid...doesn't that kind of defeat the purpose of having a relay?
