How did Gemini get its name?
Gemini - "Gemini" is the Latin translation of the Greek Didymoi, "the Twins." Many potential candidates were suggested as Gemini's original twins, but many ancient sources attribute them to being Castor and Pollux.
In Greek mythology, Pollux and Castor were twin brothers. They were the twin sons of a mortal woman Leda – however they had different fathers. Pollux was a son of the chief Olympian god Zeus, and thus Pollux was immortal and would live forever. Castor, however, was the son of a mortal man, and so he could and would die. Their love and devotion for one another was legendary, and they never separated from one another.
The two brothers, Pollux and Castor, took part in many adventures (i.e. they were involved in the Jason and Argonauts Golden Fleece thing).
The story goes that one day in battle, Castor (the mortal brother) was about to die. So Pollux asked his father, Zeus, for the permission to die along side of Castor. It was Pollux's great desire that the brothers might remain united even in death.
Instead of allowing Pollux to die, the god Zeus decided that the two brothers would now share in the immortality of Pollux and remain together forever.
However, in order for this to happen, Pollux and Castor would have to forever alternate between the light and the dark. For the rest of eternity, the brothers would spend one day on Olympus and then the next day in Hades. (Note: there are other versions and variants as to the logistics of exactly how this split took place.)