Pulse ginger in a food processor to a coarse paste (or finely grate). Bring ginger and 6 cups water to a boil in a medium saucepan; reduce heat and simmer until reduced to 3 cups, 30–40 minutes.
Strain into a large jar and mix in lemon juice and agave; add more lemon juice or agave, if desired. Let cool; cover and chill.
Do Ahead: Brew can be made 2 weeks ahead. Keep chilled.
