Who is the River woman1?
Dialogue with Jenny suggests that she's not really a "woman", but rather a sort of hallucination; Jenny's dialogue suggests that "she" represents something the observer wants, but can't have:
Jenny (matter-of-factly) It's not a woman, it's a fish. No one ever catches her.
Given the day he's had so far, Edward isn't inclined to follow up on the issue. He starts to wade back to the bank.
Jenny (cont'd) Fish looks diff'rent to diff'rent people. My daddy said it looked like the coon dog he had when he was a kid, back from the dead.
Big Fish (2003)
The idea of the "uncatchable lady fish" crops up a few times throughout the movie, most often in connection with Edward's wife Sandra; in his death scene, Edward himself draws a comparison between the two women:
But one face is missing from the crowd -- Sandra. Will turns to see she's already standing in the river beside them.
The reflection of the light off the water gives Sandra an unearthly glow. She's more tranquil and more beautiful than we've ever seen her.
Edward My girl in the river.
Big Fish (2003)
What does the River Woman represent?
It varies, depending on when she appears. In his blog John August, who wrote the screenplay, says that in Spectre she represents Edward's sexual awakening (emphasis mine):
Spectre is supposed to be a lot of different things at once. It’s the mythical town that Edward was hoping to find, but he found it too quickly. It’s a poor Southern town subject to liens and bankruptcy, which only Edward can save. It’s the location of Edward’s sexual awakening (the girl in the river) and his near-affair (with Jenny Hill). If anything, it’s a beautiful trap that Edward stumbles into twice.
johnaugust.com "Metaphors in Big Fish" February 4, 2004
In another post, August has a transcript of his first meeting with Daniel Wallace, author of Big Fish: A Novel of Mythic Proportions, on which the movie was based. According to Wallace, the River Woman represents Edward's sexual awakening, but in some contexts represents an aspect of his youth, and serves as a divine protector:
[In Spectre] I was using her as a sexual initiation, in a way, without the sex, or this awareness of the other sex, whatever. And she wasn’t going to appear again in the book. I didn’t think of her as being a reappearing character. But she came back when he was leaving Ashland [Ashton, in the movie]. When he went down to the lake, it seemed to make sense that she would be there to say goodbye to him, because she represented part of his youth — a very simple, unadorned, beautiful part — and he was leaving all that behind. And then later she comes in when the ship is sinking, and saves his life. I actually did research to find out how a boat sinks, what happens and everything. I knew he jumped overboard and went underwater, and there would be all this oil on the surface, and people get sucked under by the boat going down, and fire, and you didn't want to get oil all over you because that would make you sink. So once he was underwater, she just came back. Then, Greek myths, heroes, usually have a goddess that watches over them and protects them, like Athena was Odysseus’ protector. I think it was Athena. You know how it goes, some of the Gods are against them, and there’s always one that is for them and saves them all the time, and that was sort of a take on that.
johnaugust.com "Transcript of my first meeting with Daniel Wallace" September 9 2004
1 I'm calling her this over "Naked lady in the river" because that's how actress Bevin Kaye is credited for the role.