Following the Wikipedia citations we can see that in Life of Washington Irving, pgs 208-209, Irving did his research at both D.C. and Mount Vernon.  Where did he get his information from about George Washington?  That seems to be the key issue, since it's the earliest reference to this, and only some 50 years or so after GW's death.  I wonder if any other Washington biographers have come across Irving's source for this. 

First of all, I disagree that this is the key issue, since we have a first hand, eyewitness account in French from Comte de Moustier that GW did not append so help me God.  Plus we have other first hand, eyewitness accounts of the oath ceremony that detail what was done and said after the oath was recited and none of them mention shmG.  There is simply no reason to give Washington Irving's account precedence over the conflicting French Minister's account or over the other contemporaneous eyewitness accounts written by adult witnesses who stood on the balcony with GW.  Washington Irving's account came over 60 years after the event, which is long after everyone who is known to have stood on that balcony with George Washington was dead.  People didn't live to be 90 years old in those days.  Rufus Griswold said of Washington Irving "He had watched the procession till the President entered Federal Hall, and from the corner of New street and Wall street had observed the subsequent proceedings in the balcony." Washington Irving was six years old at the time of George Washington's inauguration. The corner of New Street and Wall Street, where Washington Irving is also identified as claiming he witnessed the inaugural in Century Magazine, volume 37, issue 6, April 1889, page 828, footnote 1 "The Inauguration of Washington" by Winthrop Clarence Bowen, is about 200 feet away from Federal Hall.Secondly, Washington Irving doesn't say where he got his information about this detail.  The reason that the other historical details of the inauguration appear to be mostly (not 100%) correct is that they were based on the previously published biography of George Washington by Mr. Sparks, from George Washington's manuscripts in the Department of State, and apparantly copied (without acknowledgement or permission) from Memoir of the life of Eliza S. M. Quincy, ed. E S Quincy, Boston [Printed by J. Wilson] 1861. But Mr. Sparks' biography, George Washington's manuscripts, and Eliza Quincy's Memoir all fail to support the claim that GW appended shmG.  Thirdly, the Senate Historical Office "so help me God" video is saying that "everyone" has followed GW in appending that phrase to the presidential oath of office.  They put out that video in March 2007, which was six months after Ray Soller corresponded with Donald A. Ritchie, Associate Historian at the Senate Historical Office, about the lack of primary source documentation showing that George Washington appended that phrase (August 28, 2006).  The Senate Historical Office claim in the video that "everyone" appended "so help me God" is just complete nonsense for which there is no evidence whatsover.  We know its false because of the Hoover audio recording (which also contradicts the JCCIC web site "Facts and Firsts" claim that Hoover affirmed his oath of office) and also because there appears to be no pre-Civil War claim of any other president adding those words to his oath of office (despite many eyewitness accounts).  To put it crudely, the Senate Historical Office is pulling this out of their rear-end.  The Senate Historical Office refuses to provide citations for their 'everyone said those words' claim.  Matthew McGowan, Professional staff for the Senate Rules Committee, told me earlier this week that the Senate Historical Office works for the Senate, not the public, and they have no obligation to provide supporting citations to the public.

Well if it is, then it would hardly be the first time that history was used as a political tool.  But perhaps your investigation should be directed at where the claims that the phrase has been attached originated.  Do these claims also date to the 19th century?

Yes.  The first known claim that a president (in this case George Washington) appended that phrase to the oath of office originated with Washington Irving in the mid-1850's, over 60 years after the fact, and was first published by the Reverend Griswold who was a friend of Washington Irving.  Washington Irving doesn't say how he came upon this detail.  Again, the Senate Historical Office is claiming that all presidents (following George Washington's precedent) appended that phrase in the "so help me God" video on the Joint Congressional Committee for Inaugural Ceremonies web site.  But George Washington couldn't have set a precedent for appending that phrase if the only known contemporaneous eyewitness account that quoted the oath as recited is correct.  That is the account of the French foreign minister who stood on the balcony with George Washington and immediately reported the details of the oath recitation back to France in writing.  According to Comte de Moustier, GW did not append "So help me God".  Furthermore, no other contemporaneous eyewitness account of the oath recitation, including the account of Georege Washington's personal secretary Tobias Lear, the secretary of the Senate Samuel Otis, of Congress, of etc. depicts him appending that phrase.Again, as I said earlier, the other claims that a president appended that phrase first appear during the Civil War.  Confederate Jefferson Davis was first.  Arthur Chester may have been the first U.S. president to append that phrase.  Arthur Chester as vice president would have taken the Civil War Iron Clad oath that had that phrase, so when he became president he may have just followed what he did when he recited the V.P. oath.  The newspapers widely reported that he appended that phrase.