As per copyright, it would be really useful to have the context, but as a guess I would say it's literally a 'barrel' of 'cockles'. So a round wooden container (barrel) containing, or made specifically for a type of shellfish (cockles.)
Leadenhall is a market in the City of London, and quite likely it would have been there in Cromwell's time. So I'm guessing Dick was a cockle seller, and his stall or stand would be called a cockle-barrel. Just a guess.