Green’s Dictionary of Slang

1. a (travelling) prostitute.

1823

‘Jon Bee’ Dict. of the Turf, the Ring, the Chase, etc. 204: Cousin-betty — a travelling prostitute.

2. a foolish woman.

1828		W. Carr Dialect of Craven I 89: Cousin-Betty. A deranged woman.
1863		E. Gaskell Sylvia’s Lovers I 304: I dunnot think there’s a man living – or dead, for that matter – as can say Fosters wronged him of a penny, or gave short measure to a child or a Cousin Betty.