Alexa Ray Joel is his daughter's name (the Ray for Ray Charles, if I'm not mistaken). Likely the ship is named after his daughter to emphasize the fact that the fishermen are indeed doing this and sacrificing themselves for their families, and to provide an element of personal feeling to the song. Joel is, as RimRod implied, very passionate about this political concern.

It's a very morose, mournful song, with gobs of traditional 80s pop rock melodrama. Much of the album is like this, and it's what makes it maybe his second best in my book (behind The Stranger).