Which aquatic plant has reddish-brown stems that are leafy, with flat leaves arranged oppositely or in whorls of three?

The correct choice is the Southern naiad. This aquatic plant is characterized by its distinctive reddish-brown stems and leafy structure, which can be easily recognized by the arrangement of its leaves. In Southern naiad, the leaves can be arranged oppositely or whorled in groups of three, adding to its unique identification features.

Sago pondweed, while also an aquatic plant, typically has a different morphology with more elongated, strap-like leaves rather than the flat leaves associated with the Southern naiad. Coontail does not display the reddish-brown stem coloration and has a different leaf arrangement; its leaves are usually more feathery and appear in a more horential structure. Filamentous algae, on the other hand, is not a vascular plant with stems and leaves but rather consists of long, thread-like structures, lacking the defined features described in the question.

Southern naiad's specific leaf arrangements and stem coloration make it the correct answer, distinguishing it clearly from the other options presented.