Dear Doc Hammer -
I have cycled six times cross country - twice in the fall - including my very first trip. That first trip, however, started at the beginning of September, not the end. I have lived in Wyoming 15 of the past 18 years and know the region well.
Is a fall trip possible this late? Barely. And it would be far better if your son had touring experience. On my first trip, I had delightful weather all September and into the first days of October, but I had a snowstorm in early October in Lincoln, Nebraska. It was only wet snow in Nebraska, but I am sure that it was serious in the higher elevations of the Rockies.
Here is a website that has climate averages of thousands of locations in the West. But remember - these are averages. It can and will be much warmer some days and much colder on others.
http://www.wrcc.dri.edu/Climsum.html
For example, Yellowstone Lake has an average high of only 49 in mid October and an average low of 22. Plus October sees an average of 8 inches of snow. But it can be as warm as 72 and as cold as 4. Plus snow is a near certainty.
It is possible if your son went via Burns thru eastern Oregon, lower elevations in Idaho via the Snake River Plains, the lowest possible crossing of Wyoming via Kemmerer and Casper, and then across Nebraska via the Platte River Valley to do it; however, the days will be getting rapidly shorter and the conditions will be deteriorating. I do not think it would be a very pleasant experience.
It is not a trip for a first-timer.