If you have a food processor, process the ginger until it is in tiny pieces. Or, hand chop it into fine pieces. You should have about 1/4 cup. It is better too much ginger than too little.
Put the ginger and 1/4 cup of the sugar in a small skillet and cook over medium heat until the sugar has melted and the mixture is hot. Do not walk away from the pan - the sugar will melt quite quickly. Remove from the heat and let the mixture cool completely.
Put the lemon zest and 3 tablespoons sugar in the food processor and process until the peel is in small bits. Or, sprinkle the sugar over the zest and chop by hand. Add the zest and sugar mix to the ginger mixture and set aside.
In a mixing bowl, beat the butter slightly, then 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 and beat until smooth. Add the lemon-ginger mixture and mix well.
Spoon the batter into the muffin cups so that each cup is about 3/4 full. Bake 15-20 minutes. Serve warm.