Well, it seems that Hardy doesn't actually know what the bluebell is either, and neither does Lee, or so he claims. Are either of them lying? What could they possibly represent?!
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I looked up bluebell + meaning and got the answer "humility"and "gratitude".
The page also cites that they're also known Dead Man's bells because fairies were believed to cast spells on those who dare to pick or damage bluebells. The people of Scotland are fond enough of the flower to continue this tradition in the hopes of protecting the little flower.
I find it interesting Scotland figures into the symbolism and Hardy's Scottish. I know the flowers were sent to Claire, but was the sender also trying to communicate something about Hardy -- or maybe Hardy's ex-wife if she too is Scottish.
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I think we all assumed that Lee Ashworth sent Claire the bluebells but it is apparent that he didn't. However, in episode 4 of season 2, at the end of the episode, after Olly posts his story that Lee Ashworth, suspected child killer in Sandbrook, is now living in Broadchurch. It was significant to notice that in the scene with the father of the murdered girl, and uncle to the other girl whose body has not been found (Guilespe?) in Sandbrook, sat in his workplace, a photograph of bluebells in fields, is on the shelves behind him. The same Guilespe? that Lee Ashworth lived next door to, whom he begs DCI Hardy to take a closer look at. Why is he sending a dried bluebell to Ashworth's missus? Does she know more than what she is letting on? Probably! And why does he not want the Sandbrook case reopened?