Rinse lemon grass; trim and discard stem ends and leaves, and pull off and discard coarse outer layer. Cut stalks into 1/2-inch pieces.
In a 2- to 3-quart pan over high heat, combine lemon grass, ginger, and broth. Bring to a boil, cover, reduce heat, and simmer for 20 minutes. Quick-chill soup (nest soup-filled pan in ice water and stir often until soup is cool, 15 to 20 minutes), or cover and refrigerate until cold, at least 1 hour or up to 1 day.
Meanwhile, in a 1- to 1 1/2-quart pan, bring 2 cups water to a boil. Add rice noodles and stir to separate. Cook until tender to bite, about 3 minutes. Pour into a fine strainer over the sink and rinse with cold water until cool, about 1 minute. With scissors, cut noodles in 2 or 3 places to make smaller pieces.
In a bowl, mix noodles, carrot, and daikon.
Pour broth mixture through a fine strainer into four wide bowls; discard residue. Mound noodle mixture equally in centers of bowls. Scatter cilantro and mint on top. Add fish sauce to taste.