What was the significance of the necklace?

I'm going to attribute it to the most powerful force in the universe -- Plot. It's a way to represent communication between Leonidas and Gorgo as well as a mechanism to 'pass the baton' to his son.

I shall further speculate that it was worn as a braid rather than a necklace so as not to interfere with visual of a ripped and oily Gerard Butler jumping and slashing all about.

While it would be awesome if it were some sort of manifestation of a Spartan ritual, the movie (even more so than the graphic novel) is so far afield from actual Spartan history, it's a surprise they still called them Spartans.