Symbolism, Imagery, Allegory
In Act 1, Edward steals Betty's necklace because he wants to give it as a gift to Harry Bagley (remember Edward has had sexual relations with dirty old Harry). Harry admits, "It's beautiful" but tells Edward, "you'll have to put it back" (1.2.253). Edward's attempt to give a gift to Harry is what some people would consider a traditional gesture of masculinity. The problem is that Edward is a little boy and he's giving the necklace to an old male pedophile instead of a woman, which complicates the idea of this being a traditional gesture of masculinity. Instead, it's a symbol of Edward's interest in pursuing Harry's (gross) affections.
Later in Act 1, Edward accuses Joshua of stealing Betty's necklace. But Harry decides to own up and calls Edward out on being a thief and a liar. Once the truth comes out, Edward tells his father that he "was minding [the necklace] for mama because of the troubles" in the nearby village (1.4.272-273). He basically tells Clive exactly what Clive wants to hear, and Clive is more than happy to accept the lie as true. This is per usual for Clive, who likes to play the whole head-in-the-sand ostrich game when it comes to avoiding the clues that his son might not be super-butch. The necklace here symbolizes Clive's ostrich-like tendencies.
The last time we see Betty's necklace is in Act 2, where we see Betty wearing it and offering it to the little girl Cathy to try on. And in case we missed the symbolic point, Cathy's mom Lin says directly to the audience, "It is the necklace from Act I" (2.1.219). Churchill isn't big on the whole subtlety thing, in case you hadn't noticed.
In this instance, the necklace becomes a symbol of the dainty femininity that Betty tries to instill into the young Cathy. But as we find out, Cathy's not about to turn into a Barbie doll just because she has tried on a necklace. She still loves to play with toy guns and do tons of typically boyish stuff.