In the late 1800s in America, people used a "buck knife" (with a handle made of an buck antler) as a dealer marker while playing poker. So the "buck" was literally passed to the next person responsible for dealing. And the "buck stopped" at the dealer.
Some think that this is where "buck" became a synonym for a dollar, considering that people eventually began using silver dollars as dealer markers. So the "buck" became the silver dollar, and hence the nickname.