Ah, so its not a Sargassum species algae, but rather a species of macroalgae or seaweed, an unknown one. The Halophila are seagrasses, and I don't know of anyone who has had them invade from seeds into their reef tanks. They are typically purposely introduced and then greatly coddled to ensure they thrive.
It is most likely you have a species of Caulerpa, a green seaweed or macroalgae. Caulerpa brachypus is very small in size, but has small serrated leaves, about 1-2" tall. The larger version of this is Caulerpa prolifera, or blade Caulerpa, which is typically 4-6" in height, long blades that are dark green.
I can see now why you've had trouble removing it as the Caulerpas have very fuzzy rhizoid material, or roots, which really latch onto porous substrate.
It is important to try to keep the population or overall biomass of this algae down. If it reaches a point where it runs out of nutrients it may sporulate (or go sexual) and release the spores, toxins and other things from the cell into the tank. Enough of this can kill or damage the fish.
Do you have any herbivores in this system? There is only one specific predator of Caulerpa that I know of, it is a species of sacoglassan (sp?) nudibranch, and are not very common in the hobby. Some tangs and rabbitfish will eat Caulerpa with relish though.
>Sarah