Make the relish by mixing all the ingredients in a bowl with a little salt and pepper.
Twist and remove the two large claws and spider-like legs from the crab and set aside. With the crab upside down, pull away the ball-like, spongy lungs. Check that the small sac and any green matter have been removed, then scoop the brown meat and skin from under the shell on to a plate. Break up the crab meat with a spoon.
Put the crab claws into a plastic bag and hit once or twice with a rolling pin to break the shells. Then, working on one claw at a time, peel away the shell, removing the white flesh with a small knife and a skewer. Add to the brown crab meat.
When ready to serve, spoon the crab into the lettuce leaves and top with spoonfuls of the cucumber relish. Roll up and eat with your fingers.