"trivial, old-fashioned, worthless," 1913 (from 1912 as a noun, "antiquated or worthless object"), said to be carnival slang and imitative of the sound of banjo music at parades [Barnhart]; compare ricky-tick "old-fashioned jazz" (1938). But early records suggest otherwise unless there are two words. The earliest senses seem to be as a noun, "maltreatment," especially robbery:
So I felt and saw that I was robbed and I went to look after an officer. I found an officer on the corner of Twenty-fifth street and Sixth avenue. I said, "Officer, I have got the rinky-dink." He knew what it meant all right. He said, "Where? Down at that wench house?" I said, "I guess that is right." [testimony dated New York August 9, 1899, published 1900]
And this chorus from the "Yale Literary Magazine," Feb. 1896:
Rinky dinky, rinky dink,
Stand him up for another drink.