The MSM has taken the position that there is no rhyme or reason for these countries. Trump advocates say it makes perfect sense as it is the list Obama used. But that, of course doesn't really answer the question. Why did Obama pick these seven? We are not privy to that rationale and therefore we have to speculate.
Other than Iran, most these countries are failed states. But there are failed states, such as Afghanistan, which aren't on the list.
It's not that they are Muslim because most Muslim majority countries aren't on the list. It's not that they are Arab because three of them aren't Arab.
The general claim seems to be that they represent a significant risk because they have some combination of state antagonism to the US, are failed states, and/or are home to active Al Qaeda/ISIS related groups.
Opponents to the ban have made the reasonable argument that only one attack on US soil has been attributable to a terrorist from one of these countries (a Somalian at Ohio State University.) But that reasonable argument is insufficient. What we really want to know is not the number of completed attacks but the number of thwarted attacks. I haven't seen anyone, Democrats or the Administration, provide that evidence.
I just came across the answer in, of all places, The Seattle Times. A perfectly fine paper, I am sure, but not one I routinely access. From Fact check: No arrests from 7 nations in travel ban? Judge in Seattle was wrong we get in the final paragraph:
Charles Kurzman, a sociology professor at the University of North Carolina-Chapel Hill, says his research shows no Americans have been killed in the U.S. at the hands of people from the seven countries — Iran, Iraq, Libya, Somalia, Syria, Sudan and Yemen — since Sept. 11. But it’s not quite right to say no one from those nations has been arrested or accused in an extremist-related plot while living in the U.S.There's a number we can work with. Those seven countries represent some 13% of the population of all countries with some affiliates of Al Qaeda/ISIS or related terror groups. Those seven countries also represent 13% of all legal immigrants/naturalizations to the country (86,000/654,000).
All told, Kurzman said, 23 percent of Muslim Americans involved with extremist plots since Sept. 11 had family backgrounds from the seven countries.
13% of the population is generating 23% of the participants in extremist plots. That's not a Pareto distribution by any means but it also is a positive and disproportionate correlation.
So, yes, there is a rationale. It is not an especially strong rationale on its own but it is perfectly valid. We are still left with the mystery of why Afghanistan isn't on the list. And even more critically, why these seven when the correlation isn't huge.