In any case, I came across something quite interesting, from 1753, just four years after the alleged execution of Avraham ben Avraham:
(From v.22, The London Magazine, Or, Gentleman's Monthly Intelligencer)
I'm not suggesting that this is in any way related; I am suggesting that this is interesting. Here is a summary of the historical evidence, or lack of, for אברהם בן אברהם.
Edit (07/01/09). Here is, incidentally, a mention of the story from 1824:

The Rabbi Mendel in question is Rabbi Menachem Mendel of Shklov, one of the original Perushim in Eretz Yisrael, and a talmid of the Vilna Gaon. Although he was born in 1750, and thus after the entire incident is alleged to have occurred, it is also evident that the story is considerably older than the date it is here recorded. It certainly cannot postdate 1808, when he arrived in Eretz Yisrael, and in fact must predate the death of the Gaon (1797). In fact, to me the way it is mentioned here has all the hallmarks of a story known to R. Mendel all his life, which puts the story even closer to 1749, when it is said to have occurred.