Less dense than water and strong enough to withstand choppy seas, storms, pirates, accidents ...

It might not sink, but if it won't hold together, you can't keep people and stuff inside it. Of course, I'm not a materials expert, but this seems like a good reason our ships are made of metal.

There are many materials that are stronger than steel and lighter than water.

It's all about costs, not about technology.

Such as?

And stronger by weight or by volume? Stronger in tensile strength or by compression? In resisting flexing? Can it handle vibration and bending without cracking? Is it tough enough not to be abraded? Can it distribute stress (by giving a bit)?

The word "stronger" by itself has no meaning in engineering. You need to specify all those things I listed when describing stronger.

Like Pykrete: http://en.wikipedia.org/wiki/Pykrete

Ah - Project Habakkuk, wartime Britain at our lunatic best - 2 million ton unsinkable aircraft carriers immune to 1000kg bombs:

http://en.wikipedia.org/wiki/Project_Habakkuk

I've always assumed this gave Iain Banks the idea for the aerial battles fought over gigantic icebergs in Use of Weapons.

I think the issue here is the material being workable for a ship. The Titanic carried hundreds of people and tonnes of cargo.

The density of the material would have to make up for the weight it was carrying, plus any water it was liable to take on in case of an accident.

The entire volume of the ship (ie, the hull, the compartments, containers, etc) must have an overall lesser density than salt water. Having a less-dense hull is not enough.