Sauté the onion in water in a large saucepan, covered, for 5 minutes, without browning. Add the potatoes and enough water to cover them, and cook covered and gently for a further 5-10 minutes.
Add the beets. Bring to a simmer for 20 min., until the potatoes and beets are tender.
Puree the soup in a blender or food processor, add the lemon zest and juice, dill and seaweed.
Serve or chill until ice cold. Delicious either way.