This could be a reference to the salt trade generally, or to the Salters Company one of the ‘Twelve Great Livery Companies of the City of London’1.
Three Cuppes Lane, on the same chequer (city block), and the Three Cuppes Inn are named in reference to The Salters Company, whose emblem includes the three cups.
[...] Nuggeston Street was what is now Salt Lane, but ‘Tottlebells Street’ is a new [...]
[...] Salt Lane [...]