Why is Rabbi Akiva the symbol of the world's largest religious Zionist youth organization ("Bnei Akiva")? (maybe even the world's largest zionist organization period.)
As an example as to why Rabbi Akiva is used as the symbol of Bnei Akiva we have History
Beginnings in Israel
Bnei Akiva was established in Israel in 1929, as the youth movement of the Religious Zionist Mizrachi organisation, by Yechiel Eliash. The official founding date was set for Lag B’Omer 5689 (28th May 1929), a date associated with Rabbi Akiva and the belief in a Jewish homeland through the story of the Bar Cochba Rebellion. From small beginnings, it became a world movement, comprising over 140,000 members in over 30 countries
Since Rabbi Akiva is known for having supported Bar Cochba as well as the story how he became a great rav and for teaching the students who allowed the Jewish people to survive and continue, he is used as the symbol of the movement.
The original story of Rabbi Akiva can be found at Like Water on a Rock
It is one of the most inspirational Talmudic stories, related in Avos D'Rebbe Nosson
As expressed on Wikipedia
The term "Bnei Akiva" and its ideology relates directly to the story of Rabbi Akiva. At the age of 40 years, after growing up tending flock, he changed his ways and decided that he needed to find out the essence of the Jewish faith. He reasoned that just as something as soft as water can penetrate a solid rock and cause it to erode, so too can the Torah penetrate into himself. Chanichim (lit. students) strive to emulate Rabbi Akiva and his qualities, namely his love of God, devotion to the Torah, respect for labor, love of Israel, and his fight for its independence.