The interpretation of Uinen as 'watery weed' comes from The Etymologies (in volume V of The History of Middle-earth, where this is explicitly given as the meaning of the name (from uy- 'a long trailing weed, seaweed' and -nend 'watery'). The reference is presumably to the legend that Uinen's hair spread like seaweed throughout all the salt waters of the world.

On the other hand, the later essay Quendi and Eldar in volume XI of the same work apparently contradicts this, suggesting (but not insisting) that the name Uinen is not Elvish at all, but originated in the language of the Valar. Perhaps both interpretations hold, with Uinen starting out as a purely Valarin name, but being later adapted by the Elves to fit their conception of this Maia of the Sea.