Preheat oven to 375F. Grease muffin tins. Cut unpeeled ginger into large chunks. Process in food processor until ginger is in tiny pieces. (You should have 1/4 cup. Better to have too much than too little.) Put ginger and 1/4 cup sugar in small skillet and cook over medium heat until sugar has melted and the mixture is hot. This only takes a couple of minutes, don't leave unattended. Remove ginger mixture from stove and allow to cool. Put lemon zest and 3 tablespoons sugar in food processor and process until the lemon peel is in small bits; or chop the lemon zest and pith by hand and then add the sugar. Add the lemon mixture to the ginger. Stir and set aside. Put the butter in a mixing bowl and beat a second or two, add the remaining 1/2 cup sugar, and beat until smooth. Add the eggs and beat well. Add the buttermilk and mix until blended. Add the flour, salt and baking soda. Beat until smooth. Add the giner-lemon mixture and mix well. Spoon the batter into the muffin tins so that each cup is three-quarters full. Bake 15 to 20 minutes. Serve warm.