China (at the time) was one of the "Big Four" Allies (France was not) during World War II. (The "United Nations" originally meant the united, anti-Axis nations.) It's true that the "Big Three" were the United States, Britain, and the Soviet Union, but there were a number of much weaker, plausible "number four" states, including China, France, and Poland (the latter two were German-occupied, with large Free French and free Polish contingents). Of these, China was the strongest and most important. France was "number five," added at the end of the war. This hierarchy was put forth by U.S. President Franklin Delano Roosevelt, who planned on having China as a counterweight after the war against British and French colonialism, and prophetically foresaw China's rise to world power (although in a different form than he envisioned).
Although China wasn't very successful in World War II, it played an important role in tying up the Japanese forces, acting as the Pacific "anvil" to the Americans' "hammer." As in Europe, the Americans fought only one fourth of the Japanese army (but most of its navy), with China absorbing most of Japan's remaining power. China's capacity in this regard was demonstrated only six years after World War II, when China spearheaded the "anti-UN" (basically anti-US) efforts in Korea.
In order to win World War II, the Axis had to defeat all three of America's major allies; Britain, the Soviet Union and China. Suppose the second worst case scenario, that the Germans had conquered the British Isles (e.g. by submarine warfare) in 1944, and European Russia by the end of 1945. Then America would be the leader of "Free British" forces in India, "Free Russia" forces in Siberia, and "free China." By mid-1945, the Allies had actually recaptured the Philippines, plus parts of Indochina and modern Indonesia, and Japan's Pacific Islands. Then Eisenhower's "Normandy" invasion could have instead liberated Japanese-occupied China in 1945, in cooperation with local Chinese forces. A "United Nations" of North and South America, China, India, Siberia, Australia, and today's ASEAN nations (even if Britain, Russia, Africa and the Middle East were lost to the Germans) would probably have sufficed to wage and win a "Cold War" with the Axis. Take China out of the equation and the "Allies" lose. (This is a thesis of my unpublished World War II book, "Axis Overstretch.")