With the back of a heavy chef's knife, bruise the lemongrass stalk to expose the inside fibers and cut it into 1-inch lengths. Combine the lemongrass pieces, ginger, sugar and 1/2 cup water in a small saucepan. Bring to a simmer and cook a few minutes until the sugar dissolves. Remove from the heat and steep for 20 minutes.
Strain the syrup into a bowl, discarding the solids. Place the bowl over a larger bowl filled with ice and whisk the syrup until it cools. Combine the pineapple juice, coconut milk, rum and all of the cooled syrup in a blender and blend; it will make a nice frothy layer on top. Pour into 4 tumblers with or without ice and garnish with the mint or lemongrass.