I saw this happen once to a friend's car.
His car too would start just once in the morning. On the next start the starter would just click-click-click. He had replaced the solenoid but that didn't solve the problem.
I checked the battery connections, which were OK. I then looked again at the red (positive) cable and saw that the exposed wires were greenish. Bingo! With an evil grin (for dramatic effect) I stuck a thin screwdriver an inch down the wire and asked him to try cranking. Away it went!
He shut it off and I then peeled back the insulation to let him in on the act. Under the first inch was green corrosion and one thin strand of wire under the red insulation.
The overnight dew had let just enough moisture accumulate in the corrosion to supplement the thin strand of wire. The high starting current then evaporated the moisture, leaving the wire to its own fate. The screwdriver bypassed the corrosion and substituted for the wires.
Yes, it was a cheap screwdriver, and yes I got a good, cheap thrill from the look on his face...
We replaced the positive cable and had no further problems.
So, please look and DO touch!