In the Bristol Journal (Dec. 15, 1787) there appears a piece called Lord George Gordon Turned Jew. It contains an account of his initial interrogation by the apprehending officer, a Mr. MacManus. The account is interesting in its own right - Gordon acted with decorum toward MacManus, but told him that he has no authority over him, or any Jew - but especially one part, which perhaps shows some of the instruction in Judaism that Gordon had heard from whichever Jews it was that he heard it from - it is some lore about converts to Judaism.
That is, Gordon had explained that "Ben Bag Bag" is a notarikon for "Ben Ger, Ben Gera [sic]." Ben Bag Bag is a name that appears in the Mishnah Avot - in fact, is the source for one of its most famous teachings - - and this name certainly does seem like a pseudonym or nickname, rather than the name of an actual person.
Most commentators have assumed that he was a convert or descended from converts, and some suggested that Bag Bag does stand for "Ben Ger, Ben Giyoret," which is what Gordon says here. (A complementary version is that Ben Heh Heh means the same thing, and perhaps the same person. There are two versions of this, in one these two men are meant to be the same person, the idea is that Heh is the gematriya of Bag = 5.)
The idea is mentioned in two prominent 18th century sources, both of which were probably fairly popular in Gordon's time, R. Yaakov Emden's commentary on the Mishnah, as well as R. Yechiel Halperin's Seder Ha-dorot, which ascribes it to the Ma'arikh. At first I thought he meant R. Menachem di Lonzano, whose magnum opus Shtei Yadot contains a section called Ma'arikh. However, I think it does not appear in his book at all. I looked it up, and then I looked at another book by that title, and it does indeed appear there. This book was printed in Paris in 1629, and the author was Philip D'aquin. It may be of interest to some that his Ma'arikh Ha-ma'arakhot was dedicated to Cardinal Richelieu of Three Musketeers fame. D'aquin was a Jew named Michael (according to Siftei Yeshenim) or Mordechai according to others, who converted to Christianity and became a professor of Hebrew in Paris; much of his book actually is based on the work of the similar name by Lonzano (see intro to Arukh Hashalem; Kohut says that most of D'aquin's original material is bad. Also see Ohev Ger (p. 133) where Shadal dismisses the book as second rate, saying that it is "only an abridged compendium of the Shorashim by Radak, the Arukh, Meturgeman and Ma'arikh [by Lonzano]." He then points out an error made by Judah Leib Benzeev who conflated the Ma'arikh of Lonzano with this Ma'arikh, by D'aquin, thereby "mixing the holy and the profane." Of course, D'aquin himself wrote that the book consists of gleanings from others - "kolel leket shikcha u-fe'ah" appears on the title page - yet surely he did not credit Lonzano..
At any rate, here is the entry in this book:
Unless his source was Lonzano (and I overlooked it) then his source may have been the abbreviation dictionary by Johann Buxtorf published in 1613, or some of the Jewish sources to be mentioned below. Here is Buxtorf:
Perhaps the first time it appeared in print was in Zacuto's Sefer Yuchasin, at approximately the end of the 15th century. Zacuto writes in his entry for Ben Heh Heh that he heard that this is the same person as Ben Bag Bag, and the gematriya of Ba"g is Heh, 5. He further identifies Ben Bag Bag with Yochanan Ben Bag Bag, who is mentioned in the Talmud, and says that he heard that Ben Bag Bag means "Ben Ger, Ben Giyoret. Another16th century source, the Midrash Shmuel on Avot, quotes this explanation in the name of R. Joseph ibn Nahmias (although often this is quoted in the name of the famous "Some Say"). Midrash Shmuel continues and cites the Rashbam, who says it was a pseudonym meant to protect the convert from persecution. which is dismissed by R. Yaakov Emden because he too assumes that Yochanan Ben Bag Bag was the same person (rather than, say, the son of this person or persons using the pseudonym). In addition to the Rashbam, Tosafot Chagigah 9b “Bar Heh Heh Le-Hillel” quotes Some Say as explaining that this mysteriously named person was a convert, and the Heh alludes to the letters Heh added to Abram and Sarai’s names and Ben Bag Bag too apparently because of the gematriya. If memory serves, the Machzor Vitry also mentions the Avraham/Sarah connection to Bar Heh Heh, firmly establishing this in Ashkenazic rabbinic tradition.
Incidentally, this whole matter is also discussed in a learned footnote in an article by Y. S. Spiegel (Yeshurun 10) and in his opinion D'aquin's book is packed with exceedingly strange explanations for roshei tevot. Spiegel also calls attention to the endorsement of D'aquin's book by the Pri Megadim.
Finally, I'd like to call attention to the Tamudic term דיירא בר דיירתא - stranger/convert son of a convert, although I'm not sure of the significance - or lack of - this yet.
Getting back to Gordon, I do think it is possible that he heard this from one of his Jewish friends, but I also found it to be not exactly common, but not completely obscure, knowledge in non-Jewish sources in the 18th century, so it is also possible that Gordon discovered this out of his own understandable interest in converting to Judaism. The fact that he also relates it to Paul's "Hebrew of the Hebrews," explaining it to mean "Jewish on both sides" as opposed to Hebrew (from one parent) needn't mean that he had not heard this from Jews. I would argue that this is exactly the sort of thing a learned 18th century English Jew would say.
Here is the whole Gordon article (click or right-click to enlarge and read):