In more than one scene of Battleship, various officers are wearing some sort of necklace, apparently permissible additions to the uniform. They are of several different designs. What are they, and what do they represent?
That thing which you call a "necklace" is actually a Lei, which is sort of a garland or wreath.
A lei is any series of objects strung together with the intent to be worn. The most popular concept of a lei in Hawaiian culture is a wreath of flowers draped around the neck presented upon arriving or leaving as a symbol of affection. This concept was popularized through tourism between the Hawaiian Islands and the continental United States in the 19th and 20th centuries.
If you remember, the Naval war games were being conducted off the Hawaiian coast.