I'd definately second The Mermaid as I can remember singing it with my brothers when I was a child and it was a huge favourite.

Other songs that I can remember singing and really enjoying included
The Raggle-Taggle Gypsies
The Streets of Laredo
High Barbary

and one that my two daughter loves now is Froggy Went A-Courting

and I think a lot of sea-shanties work because of the dynamic rhythm and energy you can put into them, plus they are often easy to pick up.