The question could seems stupid, but I think that best moment to launch a rocket is when temperature is lowest in order to achieve the best performance in a reaction engine. I do not know if there is some explanation to this or is just because that performance degradation is irrelevant.
There were several factors in the launch date
- The late John F. Kennedy had promised to go to the moon within the decade. This was a secondary concern, but an important one nevertheless
- The failure of the Soviet N1 rocket on July 3, 1969. The Soviet program was desperately trying to keep up with the American program. NASA feared the Soviets would overtake them if they slowed down.
- The Apollo Program was at peak capacity. Apollo 4 launched Nov 9, 1967. Between then and Apollo 13 on Apr 11, 1970 (30 months), NASA would launch an impressive 10 Saturn rockets (1B and V), 6 of which went to the moon in some fashion
In short, temperature does not seem to have been a concern. Apollo rockets were launched in all seasons.
The season when the Saturn V launches doesn't matter, especially in Florida where the temperature remains relatively warm year round. The Saturn V was engineered to be very resilient and to withstand almost every weather condition. You can see the launch commit criteria of modern rockets here. As Uwe said, there have been Apollo missions launched in the winter months. Apollo 8 and Apollo 17 were launched in December and Apollo 5 and Apollo 14 were launched in January (source). Furthermore, the tanks of the cryogenic fuel such as Liquid Oxygen and Liquid Hydrogen had to be good insulators because they had to remain below the boiling points (-180 °C for Oxygen and -253 °C for Hydrogen). This Apollo Flight Journal discusses the S-II (and S-IVB) stage insulation as it had J-2 engines which used cryogenic fuel. The fuel tanks for RP-1 used on the first stage (S-IC) had to have good insulation as well because it had to remain at room temperature. In some cases, rockets chilled the RP-1 to increase its density. So, ambient temperature does not affect the interior temperature of the fuel as they are excellently insulated, hence it does not affect the rocket's performance. Rockets can even launch in snow and the interior temperature of the fuel will still remain the same.
There are also other reasons why they chose July. This answer states that the opportunity to attempt a lunar landing was limited to about 16 hours every 29.5 days as the Sun would be in the right angle to see any topographical features. This excerpt is from this source.
Because lunar sunlight incidence changed about 0.5° per hour, the sun elevation angle restriction established a 16-hour period, which occurred every 29.5 days, when landing at a given site could be attempted.
So a lunar landing can be attempted about once every month. Which ties into another important factor - time. In 1962, President Kennedy in his famous speech, promised that the U.S. would land men on the lunar surface before the decade ends. Apollo 11 landed on July 20, 1969. If anything were to happen to Apollo 11 that would've caused the mission to fail, there still would've been time to land another mission on the moon before 1970 and still fulfil Kennedy's promise.