Replace the starter relay.
If in doubt, next time it's "cold" and you get the "click", instead of continueing to push the start button take a screwdriver and arc across the relay posts... bet it'll crank immediately when you arc across them! Starter relays don't last forever, the posts internally char over time (I took my last one apart to see why it wasn't working reliably, there was black charring on the posts inside the relay casing).
It may well be that your battery voltage drops a little overnight (or over the course of a week) and thus when cold there's not quite enough voltage to push thru the char inside the relay, but once you've got the engine running for a bit it tops off the battery and then has no problem forcing voltage thru the relay. Still, I bet a new relay will resolve this problem for years to come!
- Michael