Friday in this riddle is the name of a conveyance: a horse, car, motorcycle, etc.—anything that can be ridden.
The other clue word in the phrasing of the question is “on.” The riddle could be worded “came into town on,” “arrived in town on,” or even “went to town on.”
Since we pair “ride” and “on” in English, the riddle will work with anything that can be sat on and ridden. Traditionally, the answer is that the man’s horse is named Friday. While Friday is an unusual name, it is not unheard of. Other examples of English names that refer to more than one thing include April, May, and Robin.
Also, the length of time the cowboy (or man, or woman) stays in the town can vary to anything short of a week (if the cowboy stayed one week, it would be Friday again, so he would of course leave on a Friday).
As many of the other educators have astutely pointed out, the answer to this riddle is held in paying attention to the way that it is phrased. The cowboy did not drive into town; he rode into town. Thus, we can conclude that the cowboy rode in on a horse, and that that horse's name was Friday! The riddle ultimately has nothing to do with the days of the week because the cowboy can ride in and out of town on Friday the horse any day of the week.
You might be interested to know as a little side note that the word "riddle" has the same origin as the word "read." Both words stem from the Old English word "ræ̅dan," which means to "interpret or guess." Thus, that is exactly what we are meant to do with riddles— read for interpretation rather than literally.
It's a fun riddle! Obviously, the horse's name is Friday. I love riddles like this. My favorite one is the one where the animal has four legs in the morning, two in mid day and three in the evening. It's a person! Crawling, walking, then walking with a cane.
No doubt the answer to this aged riddle is the horse is named Friday. What about this riddle, "Brothers and sisters I have none, but this man's father is my father's son." Who am I? Or what about this one? Backward I am heavy, but forward I am not?
I would say pohnpei has hit upon the correct answer to this riddle. The cowboy has indeed rode into town on his horse Friday and also rode out on his horse Friday three days later.
As far as I am concerned, the only possible answer to this question is that the word "Friday" does not refer to the day of the week. There is no way that you can have three days between one Friday and the next.
So you have to think about the fact that this guy is a cowboy. If he rides into town, it's on a horse, not in a car. And horses tend to have names given to them by their owners. So it has to be that his horse was named Friday. That way, he could ride in and out of town on Friday no matter how many days he stayed.
Here's a link to that same riddle: