Jordan Should Be Arab-Palestinian

The argument related to this is multi-faceted. I will explain each component, and then explain the counterargument.

Jordan is Arab-Palestinian-Populated

Jordan, as a result of Arab-Palestinian refugees from 1948 staying in the country, has had a large Palestinian population. In fact, some estimates have placed the Palestinian proportion of the population at over 50% in the past (and possibly present). This would make the state "ethnically" Arab-Palestinian (assuming Palestinian is an ethnic group, which is actually a quite complicated question that scholars argue over), and as such the claim is that Jordan is a Arab-Palestinian state by definition.

This claim is weakened by the way that Jordan is run. It is run by a king, who has never been a Palestinian himself (in the history of Jordanian kings). If the state were a democracy, Arab-Palestinians would have had a distinct chance at running the state, and thus would have been a Arab "Palestinian" state where Arab-Palestinian self-determination could be exercised. However, this has never historically been the case.

Jordan was part of Palestine

This argument rests on the pre-1920's definition of "Palestine". As a geographic region, Palestine was never administratively defined by the Ottoman Empire with distinct borders. However, it is very likely that popular consideration of what the region "Palestine" was also included at least part of Jordan. This is made somewhat clear by the British Mandate granted by the League of Nations. In Article 25, it says:

In the territories lying between the Jordan and the eastern boundary of Palestine as ultimately determined

"The Jordan" refers to the Jordan river, which means that Palestine had uncertain boundaries and thus extended into Jordan. Indeed, for around 8 months, Palestine included (in the British Mandate for Palestine) Jordan, until it was separated rather arbitrarily. The initial thinking, many argue, was that the Jewish state would be west of the Jordan river, while the Arab-Palestinian state would be east of the Jordan river. The thinking goes that this was not implemented with the change of heart engendered by the Churchill White Paper, which said that the Balfour Declaration did not entitle Jews to "Palestine", but rather to a national home within Palestine, which meant divisions of it.

Of course, whether or not Jordan was part of historic Palestine means little in the grand scheme of things to Arab-Palestinians today. Again, they seek self-determination on the basis of Arab-Palestinian nationalism, and Jordanians (who might once have been considered "Palestinian" under the British Mandate in the early 1920's, or earlier) do not consider themselves Palestinian. Palestinians have a national movement, whose goals are debated, but there's no debating that Palestinians are not subscribing to the view that their national aspirations are satisfied by Jordan's existence, and do not regard Jordan as a state of their own.

Jordan Should Be Arab-Palestinian

This argument comes primarily from some groups, among them Israeli politicians and formerly powerful Arab-Palestinian militant organizations. The king of Jordan would also show solidarity with Palestinians by claiming "Jordan is Palestine and Palestine is Jordan", which fed into this narrative. Jordan, of course, sought to control the West Bank, but formally gave up its claim before the turn of the 21st century for various reasons, including that it remained unlikely it would ever reoccupy the territory. Pan-Arabists like George Habash of the Popular Front for the Liberation of Palestine (PFLP) would often claim that "the road to Tel Aviv runs through Damascus, Baghdad, Amman, and Cairo", though the Amman phrase was quite common. The PFLP and other groups believed that Jordan, and other Arab monarchies, stood in the way of Arab-Palestinian victory over Israel, but the slogan is sometimes distorted to mean that Amman should become Arab-Palestinian-controlled to allow Palestinians to destroy Israel and "rejoin" the historic region of "Palestine".

That's a brief overview. If you want to hear the arguments that Jordan is Palestinian as written by some, you can read the claims by Raphael Israeli, in his article "Is Jordan Palestine?". You can evaluate the arguments with the basis I've given you, and Google can help explain more counterarguments and arguments about the topic. But the claim that Jordan is a Arab-Palestinian state remains fringe.