Theory #2 - "The Wobble Saw"
Lesser known than the "Eye Wobble Wobble" legend, the "Wobble Saw" origin has gained some adherents in more recent years. Quoting Fred Thompson again:
Mencken in his American Language doubts this explanation. Some credit the term to Otis of the Los Angeles Times, an avid opponent of the IWW. Some lingual difficulty seems most likely to have been behind it, for in its sense of vacillating it fits no accusation ever made against IWW, and about the only meaning of wobbly that could conceivably fit is that of "wobble-saw," a circular saw mounted askew to cut a groove wider than its own thickness.
This is also mentioned by Mark Leier and he adds:
In his fictional account of the Centralia events, Dead March, Willimantic, Connecticut: Curbstone Press, 1980, p. 8, Tom Churchill cites a Wobbly who gives the "wobble saw" derivation but claims it was a saw that "cut in both directions".
Weighing the Evidence:
In all likelihood the "wobble saw" theory is more implausible than the Chinese Restaurant theory. While Downing's letter (supporting the "Eye Wobble Wobble" theory) lacks verifiable evidence, The "Wobble Saw" legend lacks even a known original source!
All sources that support the "wobble saw" theory date from a later date than Downing's account (1911). Fred Thompson had a thorough knowledge of IWW history (since he was a part of it for so long), but he doesn't offer any hard evidence, therefore one must conclude that he was making a guess.
The historical evidence doesn't support the theory. The theory posits that the IWW had a strong presence in sawmills prior to 1911. This is generally not the case. The IWW tended to organize the unorganized, unskilled, and foreign born workers, i.e. loggers, cat skinners, and pond monkeys for example. Mill workers tended to be skilled, and much more likely to be organized by the AFL unions. The IWW and the AFL often clashed over who would organize skilled sawmill workers in the Pacific Northwest during this time, and the AFL usually prevailed. The IWW didn't have major success organizing lumber workers in large numbers until 1917 and after, which postdates Downing's account.