Pour the elderberries, water, cinnamon, cloves, and ginger into a medium pot. Bring the mixture to a boil, then lower to a simmer for 30 minutes.
Pour the cooked elderberries though a mesh strainer over a bowl. Press the elderberries with a wooden spoon to release the liquid, and discard the berries.
Wait 15 minutes for this to cool, then whisk in the honey. Pour the mixture into a glass jar, or several small jars, and store in the fridge. The syrup will last for 3 months in the fridge, or 6 months in the freezer.