The idea was first inserted into the negotiation equation by the a group of Israeli Lefties, in 2001. Here's my description of the story, from page 268 of Right to Exist: A Moral Defense of Israel's Wars:
In July 2001, 9 months into the Jerusalem Intifada and four months into the government of Ariel Sharon, a group of some two dozen intellectuals from both sides convened to build a bridge over the ruins of peace. These were all old friends who have been meeting for many years in hope of finding enough common ground to enable the politicians to pick up the torch. Back when they started, they were unpopular pariahs in their respective communities for daring to reach out to the enemy; but over years of perseverance they had managed to pull ever larger segments of their people behind them, and from eccentrics they had become mainstream. Between them there must have been many thousands of hours of dialogue. Intelligent, educated individuals, rational realists, there was not a hard-line militant among them.Nowadays everyone likes to pretend that the demand for Palestinian recognition of a Jewish State is an evil machination of Netanyahu; most pundits are ignorant, natuarally, but Israeli critics of the demand aren't. They're hypocrites. The originators of the idea were a bunch of hard-left Israeli peaceniks.
Their idea was simple: to agree on a joint declaration calling on the warring factions to desist from their insanity and return to negotiations. The peaceniks would join hands, and with their moral authority embarrass the politicians back to sanity. The Palestinians were willing to join in stating that there should be two independent states alongside one another, but the Israelis, alerted by the fiascos of Camp David and Taba to a nuance they had previously overlooked, demanded that the statement clearly say that Israel would be a Jewish State and Palestine an Arab one. The Palestinians refused. Jews, they said, are a religion, not a nationality, and neither need nor deserve their own state. They were welcome to live in Israel, but the Palestinian refugees would come back, and perhaps she would cease to be a Jewish State.
The essence however is more important than the etymology. Is it really so essential that the Palestinians accept Zionism? If they make peace with Israel, isn't that enough?
It depends what you mean by "making peace".
If, in response to a set of Israeli terms and concessions, the Palestinians renounce their claim to a right of return of the descendants of the refugees to Israel, and they sign to a permanent, final and irrevocable end of the conflict, including renouncing any claim to somehow represent the Arab citizens of Israel, and if they prove this by desisting from brainwashing and inciting their children and general populace against Israel, then I suppose the Israeli demand can be dropped. In such conditions, in which Palestine has no more interest in the internal matters of Israel than Ecuador does, or Burundi, then indeed it will be irrelevant if they recognize us as this sort of state or that.
Until that hypothetical moment, the Israeli demand is shorthand for it. It's also longhand for the idea that the goal of the peace process is not a situation that will enable future whittling away of Zionism.