I like the Earhart idea better than her a rat. I read a theory at Lostpedia that the older woman at Juliet's book club could be the actual Amelia Earhart.
Her a rat isn't much of an anagram, not even scrambled. I think they would have came up with something better. Like maybe Eastharts Aviation where Easthart would equal she's a rat.

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Doctor playing golf. Woo, boy howdy, now I've heard everything. What's next, cop eating a donut?