Eric Partridge, A Dictionary of Slang and Unconventional English, first edition (1937), which is usually very strong on British military slang, seems to favor the Brooklands aerodrome explanation, citing Ernest Weekley, An Etymological Dictionary of Modern English (1921) as his authority, but adding a starting date for this usage of "ca. 1910–1914" (which Weekley does not).
The entire quotation from Weekley cited in the question above is actually a quotation reproduced by Weekley, from Colonel C. H. Joubert de la Ferté, I.M.S., retired; and Joubert sounds—especially in his assertion that, "Any flying-man who trained at Brooklands before the war will confirm the above statement"—very much like someone with personal knowledge of the origin of a term, who nevertheless finds himself fighting against a tide of popular support for an alternative explanation.
Joubert's account receives some unexpected support from Martin Pegler, Soldiers’ Songs and Slang of the Great War (2014), which also provides the entire "music hall monologue" of "Archibald! Certainly Not." Pegler says that the music hall monologue
was made famous in 1914 by George Robey, who recited it with great emphasis on the innuendo it contained. Robey raised half a million pounds for charity during the war and received a knighthood. The term 'Archie' was adopted by the Royal Flying Corps in 1915 as vernacular for German anti-aircraft fire and was allegedly so named after a pilot named 'Biffy' Borton who sang out the chorus whenever bursts of anti-aircraft fire exploded near his machine, much to the puzzlement of his NCO observer.
But this hypothesis requires either that Colonel Joubert be mistaken when he says that the term Archibald being applied to the pressure-change phenomenon as much as four years before World War I began, or that a rather astonishing coincidence occur in the peacetime naming of Archibald and the wartime naming of Archie after an unrelated Archibald a few years later. In his glossary entry for Archie, Pegler mentions an interesting detail about the Brooklands air-training area:
Archie From 'Archibald', anti-aircraft fire. Introduced by pilots who had flown pre-war. The origin is debatable, some sources citing Archibald sewage farm near Brooklands in Surrey, which created notorious air turbulence, the same effect as the exploding shells. Other sources cite the music hall song by George Robey 'Archibald! Certainly Not', which is reproduced later in this book.
Nothing is more likely than that "Biffy" Borton—or someone like him—made the connection between the term Archie and the popular music-hall song/monologue "Archibald! Certainly Not," and made a joke of calling or singing it out after a close call during a mission. But if a sewage farm bordering Brooklands was named after an owner named Archibald, we have a logical basis for Archibald being associated with sudden changes in pressure outside an airplane—one that doesn't rely on an arbitrary tendency of youth to give proper names to inanimate objects (as Colonel Joubert's explanation does) or on the music-hall inspired hijinks of a particular pilot (as Pegler's main-text explanation does).
Unfortunately I couldn't find any contemporaneous (early 1900s) confirmation that a person named Archibald owned the sewage farm bordering Brooklands, or that the neighboring property was then known as "Archibald sewage farm." The chronology certainly seems to fit the "Archibald sewage farm" hypothesis better than it does either of the other two—especially if we credit Pegler's comment that "Archibald! Certainly Not" did not become famous in Britain until 1914. But there is yet another complication here. A post titled "Archibald, Certainly Not!: Words and Weapons No. 4," from Oxford University's English Words in War-Time blog series, provides a couple of side-notes about George Robey and the song he popularized:
‘Archibald, Certainly Not!’, first sung by Robey in 1909, continued, for instance, to form part of his on-going music-hall repertoire. Particularly interesting, however, was Robey’s own use of modern technology, in his decision to record the song in 1911.
In other words, Robey's use of "Archibald! Certainly Not" antedates—by at least a year—Partridge's time frame for the emergence of the Brooklands use of Archibald to refer to the notoriously difficult currents above the nearby sewage farm; and Robey recorded the song in 1911, suggesting that it may have been fairly well known in Britain before the war started in 1914.
The Oxford blog doesn't mention "Archibald sewage farm" at all. If such an entity existed in the period 1910–1914, it would (I think) offer the most logical and least fanciful explanation for antiaircraft Archies. But given the lack of independent confirmation of the sewage farm's name (which forces us to rely on Pegler's account without corroboration), I can't say that this explanation of Archibald is established beyond question.
