Make the syrup: Combine the honey and 0.5 cup water in a small saucepan over low heat, stirring just until dissolved, 1–2 minutes. Let cool to room temperature, then stir in the lemon juice. Refrigerate until serving.
Bag the seeds: Place the aiyu seeds in a clean, dry muslin bag or doubled cheesecloth and tie closed. Rinse your hands and bowl thoroughly and avoid any soap or oil residue.
Extract the gel: Pour the cold mineral water into a large nonreactive bowl. Submerge the seed bag and gently rub and knead it under the water for 6–8 minutes, until the water turns pale golden and feels slippery and slightly thick; keep the water cool and avoid distilled or softened water.
Set the jelly: Lift out the bag, squeeze to release the last of the gel, and discard the seeds. Leave the bowl undisturbed at room temperature until the surface is set and the jelly jiggles as one mass when tilted, 30–45 minutes. For a firmer texture, chill an additional 30 minutes.
Serve: Cut the jelly into spoonable chunks or 1-inch cubes. Divide into bowls over ice cubes and spoon the lemon-honey syrup on top to taste. Serve immediately, keeping jelly and syrup stored separately until serving to prevent weeping.