@Efi Grate or chop the ginger very very finely (ground up into a paste if possible). Boil with water and steep for maybe 10 minutes.

Strain the ginger and retain the water. Dissolve an equal quantity of sugar into the still hot water.

Congratulations, you have created a ginger-infused simply syrup.

Add an ounce to a glass of club soda, use it in place of regular simple syrup in cocktail recipes, or add it to whatever you'd like to add a little sweetness and ginger flavor.