Theory #1 - "Eye Wobble Wobble"

Also known as the "Chinese Restaurant Owner Theory", this is the most often cited and embellished theory. There exists plenty of anecdotal evidence to support this theory as having a grain of truth to it. Although it is equally likely to be little more than a cleverly crafted tall tale or yarn. We quote from Three Original Sources:
(1) The earliest known reference to the term:

In Vancouver, in 1911, we had a number of Chinese members, and one restaurant keeper would trust any member for means. He could not pronounce the letter "w" (due to the "l" sounds in the pronunciation of the letter), but called it "wobble" and would ask, "you Eye Wobble Wobble?" and when the [red] card was shown credit was unlimited. Thereafter the laughing term amongst us was "I Wobbly Wobbly".

--Mortimer Downing, IWW Member. Quoted in Jack Scott, "How the Wobblies Got their Name," in his Plunderbund and Proletariat (Vancouver, BC.: North Star Books, 1975), p. 153. Also quoted in Jerry Lembcke and William M. Tattam, One Union in Wood, A Political History of the International Woodworkers of America (New York, NY.: International Publishers and Madeira Park, BC.: Harbour Publishing, © 1984), pp. 188-89 n31.
(2) The following account is from the Official IWW History:

It was at this time (1912 during a "thousand mile picket line" railway strike in British Columbia) that the term "Wobbly" as nick-name for IWW came into use. Previously they had been called many things from International Wonder Workers to I Won't Works. The origin of the expression "Wobbly" is uncertain. Legend assigns it to the lingual difficulties of a Chinese restaurant keeper with whom arrangements had been made during this strike to feed members passing through his town. When he tried to ask "Are you I.W.W.?" it is said to have come out: "All loo eye wobble wobble?" The same situation, but in Vancouver is given as the 1911 origin of the term by Mortimer Downing in a letter quoted in Nation, Sept. 5, 1923..."

--From The IWW: Its First 100 Years by Fred W. Thompson and Jon Bekken, 2006, IWW: Cincinnati, page 60..
3) This account is further elaborated in the following quote:

The word "Wobbly", a nickname for IWW members, humorously illustrates the union's efforts to combat racism. A Chinese restaurant keeper in Vancouver in 1911 supported the union and would extend credit to members. Unable to pronounce the letter "w", he would ask if a man was in the "I Wobble Wobble". Local members jokingly referred to themselves as part of the "I Wobbly Wobbly," and by the time of the Wheatland strike of 1913, "Wobbly" had become a permanent moniker for workers who carried the red card. Mortimer Downing, a Wobbly who first explained the etymology, noted that the nickname "hints of a fine, practical internationalism, a human brotherhood based on a community of interests and of understanding."

--Mark Leier, Where the Fraser River Flows, The Industrial Workers of the World in British Columbia (Vancouver, BC.: New Star Books, 1990), page 35.

Weighing the Evidence

Conceivably, Downing's account could be the honest truth. According to Dan Cornford (in Workers and Dissent in the Redwood Empire, ©1987, Temple University Press), The IWW was the first labor union in North America to refuse to discriminate against Chinese and Chinese Americans. (Many earlier left-wing organizations, including the Greenback Labor Party and the Knights of Labor discriminated vehemently against Chinese and Japanese Americans. Former members of these organizations (such as George Speed) later joined the IWW and jettisoned their racism). Such interracial solidarity most certainly did not go unnoticed in the Chinese American community, and they would likely have responded favorably to the IWW.
However, all the evidence of the "Chinese Restaurateur Theory" apparently stems from Downing's letter. There is no known independent source that verifies Downing's story. His account may just as easily be a romanticized embellishment of the truth, or it could be pure fiction, and there is no credible proof that it isn't. Downing's narrative also suggests deeply ingrained stereotypical views of Chinese and Chinese-American speech patterns, even by 1911 standards.
Quoting Mark Leier again:

In a letter to the author, dated 31 January 1989, Craig M. Carver, managing editor of the Dictionary of American Regional English states that the Chinese restaurateur version is not given "much credence ... because the story is simply unverifiable." Those with a scientific bent must conclude that the etymology is unknown; romantics may choose to stick with Downing.

--Mark Leier, Where the Fraser River Flows, The Industrial Workers of the World in British Columbia (Vancouver, BC.: New Star Books, 1990), p 35.