There might be no reason but I doubt the date had simply no reason.
I believe there may have been personal significance to Neil Armstrong. Alternatively picking a summer date may have been safer.
The reasons for that date were:
There were a number of considerations which determined the launch windows for a lunar landing mission. These considerations included illumination conditions at launch, launch pad azimuth, translunar injection geometry, sun elevation angle at the lunar landing site, illumination conditions at Earth splashdown, and the number and location of the lunar landing sites.
The time of a lunar landing was determined by the location of the lunar landing site and by the acceptable range of sun elevation angles. The range of these angles was from 5° to 14° and in a direction from east to west. Under these conditions, visible shadows of craters would aid the crew in recognizing topographical features. When the sun angle approached the descent angle, the mean value of which was 16°, visual resolution would be degraded by a “washout” phenomenon where backward reflectance was high enough to eliminate contrast. Sun angles above the flight path were not as desirable because shadows would not be readily visible unless the sun was significantly outside the descent plane. In addition, higher sun angles (greater than 18°) could be eliminated from consideration by planning the landing one day earlier where the lighting is at least 5°. 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. The number of Earth-launch opportunities for a given lunar month was equal to the number of candidate landing sites.
The time of launch was primarily determined by the allowable variation in launch pad azimuth and by the location of the Moon at spacecraft arrival. The spacecraft had to be launched into an orbital plane that contained the position of the Moon and its antipode at spacecraft arrival. A 34° launch pad azimuth variation afforded a launch period of 4 hours 30 minutes. This period was called the “daily launch window,” the time when the direction of launch was within the required range to intercept the Moon.
Two launch windows occurred each day. One was available for a translunar injection out of Earth orbit in the vicinity of the Pacific Ocean, and the other was in the vicinity of the Atlantic Ocean. The injection opportunity over the Pacific Ocean was preferred because it usually permitted a daytime launch.
So they basically had one launch opportunity per month. I suspect they chose the earliest possible opportunity that could be achieved safely.