From Namespedia - Names Meaning and Origins
Meaning: From the Hebrew name בִנְיָמִין (Binyamin) which means "son of the south" or "son of the right hand". Benjamin in the Old Testament was the twelfth and youngest son of Jacob and the founder of one of the southern tribes of the Hebrews. This name was also borne by Benjamin Franklin, an American statesman, inventor, scientist and philosopher.
Comments: The aforementioned Benjamin Franklin was perhaps the most celebrated bearer of the name. Other illustrious bearers include Benjamin Disraeli, British Prime Minister; Benjamin Britten, British composer; and Benjamin Netanyahu, Israeli Prime Minister.
Beatrix Potter bestowed the name on Peter Rabbit's memorably naughty cousin, "Benjamin Bunny", who starred in several of her tales.
One of the more interesting variants of the name: "Benxamin", from the Galician (a Spanish dialect).
Nicknames other than the usual "Ben": Prime Minister Netanyahu is famously nicknamed "Bibi". Other, unexpected short forms of Benjamin: "Benj"; "Binns"; "Bim"; "Benk(s)" or "Benx"; and "Bank(s)", among others.