For many centuries, the ideal pub was one that you could see out of, but no-one could see into. Pubs and Inns would sometimes erect latticework in order to achieve this and call themselves the Red Lattice or Green Lattice, which became corrupted into the Lettuce. This chain's name is probably a combination that was researched and invented by a marketing agency!
Thanks to the book "Pub Signs" by Paul Corballis