Green’s Dictionary of Slang

rejection, dismissal; usu. as get/give the dinky-dink.

1890		Salt Lake Herald (UT) 9 Dec. 6/1: Thoe players who are fearful that they will be given what Al Johnson [...] calls the ‘dinky dink’.
1893		Anaconda Standard (MT) 24 Oct. 7/1: You can’t give me the dinky dink cull, I’m dead on your curves.
1898		Eve. Post (Wellington) 9 Mar. 2/7: [In New York] ‘Dinky Dink’ is slang for the cold shoulder.
1899		R. Whiteing No. 5 John Street 306: We just let it all roll by, and gave all the bounders that preached it the dinky-dink.
1901		W. Irwin Love Sonnets of a Hoodlum IV n.p.: I got the dinky dink.
1908		Dly Public Ledger (Maysville, KY) 3 June 2/2: Those persons who assert that President Roosevelt got the dinky-dink from Congress are evidently not converant with the facts.
1916		El Paso Herald (TX) 19 Aug. 19/1: Chris frankly admitted that he was the recipient of the much feared ‘dinky-dink’.