Slice ginger into chunks and place in bottom of mixing glass. Muddle until broken up well, about 1 minute. Strain juice through fine-mesh strainer, pressing on solids with the back of a spoon to extract juice.
Cocktail
In a cocktail shaker, muddle mango, cilantro, lemon juice, ginger juice, and simple syrup until the mango is completely dissolved.
Fill shaker with 2/3 full with ice and add tequila. Shake until well chilled, about 15 seconds. Using a strainer, pour into a serving glass filled with ice.