I forgot Rebekah threw the necklace away like that. I think after finding out her mother wanted to kill her she decided that she no longer cared about the necklace at all.
Last time I saw it was in a flashback on The Originals and Rebekah was wearing it (it was when they first turned/running from Mikael).
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I think the scene was just cut oddly. Caroline was on the phone with Bonnie/Enzo and something triggered her memory to want the necklace so then she ran into Stefan's bedroom and saw him dead on the floor. I thought it was because the necklace was in his room, but maybe she only went in there to ask him where the necklace would be
.