"The Big Lie" was what Lyndon Johnson called the Arab suggestion that the U.S. was directly supporting the Israelis in the Six Day War. The contingent of Arab countries (Jordan, Egypt, Syria, Iraq) who were involved in armed hostilities with the Israelis, claimed that the U.S. was bombing their bases to aid Israel, or as they preferred, "the Zionists".

The "Big Lie" put Johnson in a difficult position. The U.S. wanted to support stable regimes like Hussein in Jordan, but the U.S. could hardly assist them diplomatically during the conflict, while at the same time they accused the U.S. of attacking them in the media and on the floor of the U.N. This forced Johnson to privately, cast his lot with the Israelis since the Soviets were assisting the Arab countries (particularly Syria). Johnson, unlike Truman, was not a fan of the Israeli state and didn't want to pick sides in the Middle East, but because of the Big Lie he was forced to.

While the U.S. remained uninvolved militarily, the shift behind the scenes diplomatically, allowed the Israeli government both more time to engage in hostilities and put a check on direct Soviet involvement in the region -- the thing the IDF feared the most. The most tangible direct result of the extra time the Israelis secured was the taking of the Golan Heights in southern Syria.

Ironically, in the end the lie Arab propagandists were claiming, (diplomatically at least) came true because of the original lie.

Source: Six Days of War: June 1967 and the Making of the Modern Middle East, by Michael B. Oren