don't give/care a rap for

Worthless to me; without any interest. The “rap” in question was a base halfpenny that was worth only half a farthing and was circulating in Ireland in the early eighteenth century because small coins at that time were very scarce. Jonathan Swift described it in Drapier’s Letters (1724): “Copper halfpence or farthings . . . have been for some time very scarce and many counterfeits passed about under the name of raps.” Consequently the name was adopted for anything of little value and was so used by the early nineteenth century. W. H. Ainsworth wrote (Rookwood, 1834), “For the mare-with-three-legs [i.e., the gallows] I care not a rap.”