You guys are on the right track about pre-concentration of the sample to get a higher level to test. Here's a book that might give you an idea how big of a topic that can be.
http://books.google.com/books/about/Preconcentration_techniques_for_trace_el.html?id=FV6z2VvHaVsC
You can't do it by boiling though. While you will get the volume of water down, you will probably also put your copper into insoluble states that may fool your test.
I'm a chromatographer, so my first choice is naturally going to be a column. It works kind of like the DI resin in your water purifier, except that the media will selectively chelate to copper. There are resins made specifically for this purpose. The book linked above lists some of them. A large volume of sample is passed over the resin and the copper is trapped there. Afterwards, the copper is eluted with a small volume of highly concentrated chelating agent like EDTA.
Not only does this get your copper concentrated, but it also removes most of the other ions from the saltwater. With those gone, we can play some solubility games to get it even more concentrated or remove the chelating agents. For example, we can precipitate the copper as the hydroxide, spin it out in a centrifuge, and then redissolve in a smaller amount of nitric acid.
Unfortunately, I don't think any of that can be done at home with commonly available items. About the closest we can get is the media in the cuprisorb.
In the lab setting, copper is usually pre-concentrated on column and then analysed by AAS or ICP-MS. Obviously the average reefer doesn't have access to those machines.
There are some really interesting electrochemical methods available as well.
In this book: Methods of seawater analysis By K. Grasshoff, Manfred Ehrhardt, Klaus Kremling
http://books.google.com/books?id=ep1A02HNlMIC&printsec=frontcover#v=onepage&q&f=false
Complexation and precipitation of copper onto an electrode followed by cathodic stripping voltammetry is reported as having a limit of detection of 0.1nmol/L (6.35 ng/L or 0.00000635 ppb).
I also recently saw a paper about selective precipitation titration of copper with sulfide using a sulfide ion selective electrode. I remember it had a low LOQ.