I use seaweed in soup (yes soup, not soap) quite often, so I considered using seaweed-soaked water in place of plain distilled water, but have not as yet tried it. I have read of people using rice water and also potato water in soap, so it made sense to me to try seaweed water. One of these days I will give all of these a try.
As for bits of dried seaweed, unless it is finely finely finely ground up, as in powdery consistency, I would not be tempted because I have experienced how scratchy other organics become in soap, even when they start out wet & soggy. The dried seaweed I use is very very hard when dry, so I know it would be like little sharp knives in soap. Maybe Nori sheets would be less so, but it's not Nori that I generally use for cooking.