Engine will not start

Everything you described sounds typical of a bad connection on a battery cable or where the smaller positive wire bolts to the under-hood fuse box. Finding this should be easier since you can get it to go to the bad condition. Turn the headlights on so some current is always trying to flow, then start right on the battery bolts and measure the voltage. If you find 12.6 volts, move one meter probe from the negative post to a paint-free point on the body. You should find the same voltage. Move the probe from the positive bolt to the stud on the fuse box. You're looking for the first place the voltage is considerably lower. That will be the point of the bad connection.

If your saying, there was no starter action at first, and the clicking came from the starter relay, I'd check voltage reading at the control side of the relay. A relay may click a lot when there is not enough available voltage to keep the relay closed. You'll have to test when the problem occurs. It might be a problem with a wiring circuit or connector?