Sukkot marks the beginning of the rainy season, and Rabbi Eliezer opines that we should start saying masheev haruach umoreed hageshem starting the first day of Sukkot. While Rabbi Yehoshua does not disagree that, in theory, Sukkot is the correct starting point, "since rain on Sukkot is a sign of a curse", he ruled that we should push off the recital of masheev haruach until Shemini Azeret—which is the practice we follow today.
"Memati mazkirin gevurat geshamim, from when do we begin to mention the power of rain?" (Ta'anit 2a). Masechet Ta'anit opens with a discussion of when we are to begin reciting masheev haruach umoreed hageshem and v'ten tal umatar during davening. While most of us think of fast days in the context of either Yom Kippur or the destruction of the Temple, Masechet Ta'anit, the tractate of fasting, deals primarily with fasts due to lack of rain.