Back in the days of the Ottoman Empire, Palestine was a part of that Empire which was contested repeatedly by the Egyptians (as a puppet state of Britian) and the Ottomans, finally being claimed as officially Ottoman territory in 1840. Then when World War I broke out and the Ottomans sided with Germany and Austria-Hungary, Britian, France, and pre-Leninist Russia began making plans on how to split up the empire, though Russia later left the agreement due to the whole revolution thing. This agreement became known as Sykes-Picot and sought to divide the lands of what are now Palestine, Israel, Jordan, Syria, and Lebanon into British and French Mandates, with the French taking the northern half (Syria and Lebanon) and the British taking the Southern half and dividing it by the Jordan river, creating Palestine and Not-Palestine (Just joking it was called Transjordan, which literally means the other side of the Jordan).

None of these territories had historically existed on their own prior to this, and to top it off, in order to get the Arabs to revolt against the Ottoman's the British and French made a boat load of promises to a bunch of people that under British rule they would get freedoms and sovereignty over their own lands, these promises were made to Palestinians hoping to have their own nation as well as European Jews through the Balfour Declaration which declared the intent of the British kingdom to turn Palestine into a national home for the Jews. Under British Mandate, the Arabs revolted multiple times including in 1920 as a reaction to Zionism (Nebi Musa Riots), in 1929 over access to the Western Wall (Buraq Uprising or 1929 Massacre depending on who you are), and in 1936 as an official cry for independence and the end to open Jewish Immigration (Arab Revolt of 1936 or Great Revolt).

During World War 2 the Palestinians continued to hold out hope for independence by supporting Adolf Hitler. In fact, the Grand Mufti of Jerusalem at the time, Amin al-Husseini, openly supported and visited Hitler regularly. The hope was that Hitler's mid-East campaign would result in their independence from the British and would allow them to purge the Jews from their land.

After the war tensions reached a boiling point as the British government saw fit to ease everyone's moods by meeting halfway and prohibiting Jewish refugees from entering Palestine. So then, both the Jews and Palestinians took to open combat. The UN stepped in and ordered the creation of a Nation of Israel and a Nation of Palestine. Israel accepted this resolution, but the Arabs did not. So the Nation of Israel was created, Palestine was not, and a full out war broke out in Israel-Palestine.

So, to wrap up, the desire for a Palestinian nation started with the British Mandate and promises made during the Arab Revolt in World War I. These desires only became greater as Britain reneged on their promises to the Arabs in favor of promises made to the Jews, resulting in revolts and uprisings across the region. In the modern day, Palestinian National Identity is mostly associated with the PLO and figures like Yassir Arafat, but it existed long before them.