Chestnut is a British slang term for an old joke, often as old chestnut.
The term is also used for a piece of music in the repertoire that has grown stale or hackneyed with too much repetition.

A plausible explanation for the term given by the Oxford English Dictionary

Oxford English Dictionary

The Oxford English Dictionary , published by the Oxford University Press, is the self-styled premier dictionary of the English language. Two fully bound print editions of the OED have been published under its current name, in 1928 and 1989. The first edition was published in twelve volumes , and...

 is that it originates for a play named "The Broken Sword" by William Dimond, in which one character keeps repeating the same stories, one of them about a cork tree

Cork tree

Cork tree may refer to:* Cork Oak, the tree from which most cork is harvested* Chinese Cork Oak, a tree from which cork is occasionally harvested* Cork-tree, a species of Phellodendron...

, and is interrupted each time by another character who says: Chestnut, you mean . . . I have heard you tell the joke twenty-seven times and I am sure it was a chestnut.

The play was first performed in 1816, but the term did not come into widespread usage until the 1880s.