(idiomatic,Britain) Not to have the slightest interest or concern.
1998: Willis Hall, Keith Waterhouse, Billy Liar - He knows what he can do with his calendars. I don't give a monkey's. I'm leaving. I've got another job.
2002: "A British managing director", quoted in Robert Westwood, Stephen Linstead, The Language of Organization - I couldn't give a monkey's. What I am interested in is the things that drive the big numbers.
2004: Randall Stevenson, The Oxford English Literary History, 1960-2000 - Though a huge majority of the population may have continued not to give a monkey's for Derrida, Lacan, or Foucault themselves...
2005: Justine Roberts, Carrie Longton, Mums on Pregnancy: Trade Secrets from the Real Experts - A friend living in France reports that the French don't give a monkey's about unpasteurized dairy products (cheese, etc.) but that they are completely paranoid about le toxoplasmosis.