Combine kombu and water in a large saucepan; bring to a boil. Reduce heat and simmer 1 minute. Drain in a colander over a bowl, reserving liquid. Discard kombu. Combine 1/4 cup reserved liquid and miso, stirring with a whisk.
Heat oil in pan over medium heat. Add daikon, carrot, and onion bottoms; cook 3 minutes. Add remaining reserved liquid; bring to a boil. Reduce heat; simmer 5 minutes. Add watercress and dill; cook 1 minute. Stir in the miso mixture, and cook 1 minute. Stir in lemon juice. Sprinkle with green onion tops.