Coney-catching is Elizabethan British slang

Slang

Slang is the use of informal words and expressions that are not considered standard in the speaker's language or dialect but are considered more acceptable when used socially. Slang is often to be found in areas of the lexicon that refer to things considered taboo...

 for theft through trickery. It comes from the word "coney

Coney

-Places:* Coney Arm, a settlement in Newfoundland and Labrador* Coney Beach Pleasure Park, an amusement park in Wales* Coney's Castle, a fort in Dorset, England* Coneygar, a suburb of Bridport, Devon, England* Coneyhurst, a hamlet in West Sussex, England...

" (sometimes spelled conny), meaning a rabbit

Rabbit

Rabbits are small mammals in the family Leporidae of the order Lagomorpha, found in several parts of the world...

 raised for the table and thus tame.

A coney-catcher was a thief or con man

Confidence trick

A confidence trick is an attempt to defraud a person or group by gaining their confidence. A confidence artist is an individual working alone or in concert with others who exploits characteristics of the human psyche such as dishonesty and honesty, vanity, compassion, credulity, irresponsibility,...

.

The term was used in 1592 pamphlets by Robert Greene

Robert Greene (16th century)

Robert Greene was an English author best known for a posthumous pamphlet attributed to him, Greene's Groats-Worth of Wit, widely believed to contain a polemic attack on William Shakespeare. He was born in Norwich and attended Cambridge University, receiving a B.A. in 1580, and an M.A...

, the titles of which included "The Defence of Conny-catching," in which he argued there were worse crimes to be found among "reputable" people, and "A Disputation betweene a Hee Conny-catcher and a Shee Conny-catcher."