It's not entirely clear where this phrase originated, though Sir Arthur Conan Doyle did not write this specific, well-known phrase. From what I've found out, the quote was in a movie and in a 1929 New York Times review of the movie, but it could well have started before this and already been considered one of Sherlock Holmes' signature lines, though not penned by the author.
(The last site I list, which investigates hoaxes and rumors and questionable quotations, gives more information than the other two.)
From http://www.saidwhat.co.uk/quotes/misquotes/ about the quote "Elementary, my dear Watson" - Conan Doyle (Sherlock Holmes)
"This quote is not actually found in any of his books, but rather found in a film review in the New York Times, 19 October 1929."
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Wikiquote http://en.wikiquote.org/wiki/List_of_misquotations says:
"• This phrase was never uttered by the character in any of Sir Arthur Conan Doyle's written works. Though "Elementary," and "...my dear Watson." both do appear near the beginning of The Crooked Man (1893), it is the "...my dear Watson" that appears first, and "Elementary" is the succinct reply to Watson's exclamation a few lines of dialogue later. This is the closest these four immortal words ever appear together in the canon.
"• The association of this quote with the Sherlock Holmes character likely comes from the closing lines of the 1929 film The Return of Sherlock Holmes:
Watson: Amazing, Holmes.
Holmes: Elementary, my dear Watson, elementary."
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Among other things about this quotation, Snopes: http://www.snopes.com/quotes/signature/elementary.asp says:
"As is often the case (with literary characters), the best-known form of this signature phrase apparently originated with a derivative, non-canonical work: In the 1929 film The Return of Sherlock Holmes, written by Basil Dean and Garrett Fort, the renowned Baker Street detective (as portrayed by actor Clive Brook) speaks the line, 'Elementary, my dear Watson, elementary.' Determining whether that movie was primarily responsible for popularizing this phrase is problematic, though, as the New York Times noted in its review of the film that:
"In the final scene Dr. Watson is there with his 'Amazing Holmes,' and Holmes comes forth with his 'Elementary, my dear, Watson, elementary.'
"This suggests that 'Elementary, my dear, Watson' might already have been a familiar phrase to audiences, and the writers of The Return of Sherlock Holmes merely capitalized on that familiarity by incorporating the line into their film. The actor William Gillette is often credited with originating this phrase in a slightly longer form in the 1899 stage production Sherlock Holmes (for which he both wrote the script and played the lead role), during which he (as Holmes) reportedly uttered the line, 'Oh, this is elementary, my dear Watson.' However, a 1999 biography of Sir Arthur Conan Doyle indicates that it is unclear exactly when (if ever) Gillette actually spoke this line on stage:
"Many of Gillette's inventions and mannerisms were absorbed into the Holmes mythology. He may have been the first to utter the words 'Elementary, my dear Watson,' though the line does not appear in any published version of the script -- nor in any story by Conan Doyle."
***Note: This is, no doubt, more information than you wanted, especially considering it doesn't give a conclusive answer. One thing seems apparent: the phrase came about because the Sherlock Holmes stories were enacted. It would be interesting to know what, if anything, Sherlock Holmes' creator thought of the line.