The Oxford English Dictionary conjectures that this expression could also have derived from U.S. military slang for sycophants, "brownnosers", while mentioning the less-likely but popular etymology that derives it from the awards system of the Brownies Girl Guides/Girl Scouts. "Brownie" itself in the sense of "brown-noser" was in use in the 1940s.
The OED reports its first appearance in print as 1963, though the origins of the phrase predate this. Its frequent appearance in newspapers in the 1950s date back to the earliest known usage in 1951, where a man in the Los Angeles Times speaks of earning favor with his wife in terms of brownie points.
Now as to where to cash them in - I have found Brown Bank (undersea bank), in the Atlantic Ocean. - they will presumably liquify your assets as soon as contact them in person.