The phrase “not worth a rap” means: Having no intrinsic value; not worth a straw, nor a tinker’s dam, nor beans.
We don’t know just how rap got its name, but it was a very small coin which, though not legal tender, was passed for a halfpence in Ireland during the early eighteenth century.
It was because of the lack of legal small currency and in protest against a lopsided patent issued to one William Wood by George I for the coinage of copper halfpence in Ireland that Dean Jonathan Swift wrote the celebrated Drapier Letters in 1724.
Concerning the rap, he wrote in one of the letters, “Copper halfpence or farthings . . . have been for some time very scarce, and many counterfeits passed about under the name of raps.”
And when we say, “I don’t care a rap,” it is the same worthless coin to which we refer.