PhD application: my country isn't listed

The United States does not recognize the statehood of Palestine (meaning technically that can't be your home country as far as that government is concerned), which makes it somewhat unsurprising that an American university might not have it listed as a country.

The Department and School likely doesn't have a particular stance on the issue - they probably didn't even design the application. Of course the ideal option would be to call them, but as this isn't an option at the moment, I'd put something close (Jordan - which as you mention you hold a passport for) and then call them to clarify as soon as you are able to do so.

Also, as this is clearly a recurring problem, you should seriously consider finding out if a given school's application has this problem well before it is due.

First, this is just an error on their part, and not an issue of politics or policy. Contact them and they'll fix it, and in the meantime pick something close, as said in comments.

That said, it might be best to just pick Jordan anyway. If you were to be accepted, which passport would you use to get a visa? As a practical matter, this might be affected by which consulate you go to, which is obviously this is a little tricky since there's no consulate in the West Bank. If the answer is your Jordanian passport at the consulate in Jordan, then I'd pick Jordan as nationality, even if you currently reside in the West Bank. They're asking this question because they want to know what your visa situation would be.