Mine was turning over but wouldn't fire in the mornings, but then working just fine the rest of the day as you said. It was especially bad below freezing. I was always able to get it to fire, but only after about 10 minutes of trying. I knew I didn't have the issues with the computer that was linked above since I could hear my fuel pump turning on and I could see my check engine light turn on then off. I didn't have any codes and I also tried turning the key on and off to prime the system a few times but that didn't work.
For me, replacing the plugs, wires, cap, rotor, and coil completely solved my problem. I'm back to cranking over 2 or 3 times and it fires, even when it's cold.