Apparently, even though Elizabeth was a Unionist and opponent of slavery there was an effort to paint her as a southern sympathizer at the time of her death. The "anecdote" of son Edward possibly being a member of the conspiracy probably arises from that. I can find no mention of it before the 1890's. Elizabeth did have a son, Appleton Oaksmith who did support the Confederacy as a blockade runner, was caught and then escaped from a Boston prison in 1862 and fled to London.
Here's a link to a brief 1880 bio of Edward Oaksmith: