Blend strawberries, 1 cup water, and mint leaves in a blender until smooth, 20 to 30 seconds; strain into a 2-quart pitcher. Stir coconut water, 1 cup water, lime juice, lemon juice, and agave nectar into the strawberry mixture.
Mix chia seeds in small increments into the strawberry mixture, stirring continually to keep from clumping together. Refrigerate mixture until chia seeds fully absorb the liquid, 30 minutes to overnight.