But why did they reject Zion, if that's the name of the movement to begin with? Why pick Israel, if that's the name of the northern kingdom that got corrupted while the Kingdom of Judah was the one that was with God?
"Israel" was used to refer to the United Kingdom in Judean texts, so it's better suited to a unified Jewish state. Those righteous Judean kings are always measured up against David, after all.
Edit: Zion does make more sense, I agree.
tl;dr they were named by different people at different times
"Zion" refers to Jerusalem (well, a specific mountain in Jerusalem, and thus the whole city by metonymy). The choice of the term "Zionism" to refer to the Jewish nationalist movement is for somewhat historical reasons. Contemporary with the foundation of Zionism was the formation of a series of groups called the "Chovevei Tsion" ("Those who love Zion"). These were organizations seeking to establish Jewish settlements in Palestine as a way to get Jews away from the enormous antisemitism and pogroms of Russia; notably, they didn't care much about a Jewish state and were happy to live under Ottoman benign neglect. It was actually these groups, rather than the more abstract and intellectual "Zionists proper" of Western Europe that began the first of what we would now call Zionist settlements, including the very first one, Petach Tivka ("Opening of Hope"; the specific name is a biblical reference, but the choice should be pretty clear). The link between the Chovevei Tsion and the more "political" Zionists like Herzl who sought specifically the establishment of a Jewish state was probably Leon Pinsker, author of Auto-Emancipation, probably the first written document calling for the establishment of a Jewish state (though offering several possible sites), since he was also involved with the former. The first use of the term "Zionism" in reference to the movement was in the journal Self Emancipation. As far as I know, the reason for this choice was never stated, but the editors would certainly have read Auto-Emancipation, and likely chose it in reference and deference to Pinsker and the Chovevei Tsion.
In contrast, "the land of Israel" is the traditional Jewish term for region of Palestine/Judea/Southern Syria/Whatever. The people choosing the name weren't thinking about 2500 year old kingdoms. There's also the fact that "children of Israel" is
the traditional term for Jews, moreso than the term "Jews", actually, which technically refers only to members of the tribe of Judah slash people from the southern Kingdom of Judah (which also included some people from a few other tribes). The term Hebrew (the root of the word for Jews in many eastern European languages) also works, but Israel was preferred. This is important, because the Zionists wanted Israel to be a nation-state, so it made sense to name the state after the nation, as, I dunno, France is called France and not Gaul (the fact that the nation and the region had the same name made the choice of name something of a no-brainer).