Where does the 10% myth come from?

No one knows exactly.  A nice summary of what we do know is provided in a recent WIRED piece here. William James was thought to play a role, based on a quote from Dale Carnegie's book, How to win friends and influence people, but this may have been a misquote.  Kolb and Wishaw's classic text, Fundamentals of Human Neuropsychology suggests Flourens work in the early 1800s as a likely empirical foundation for the myth.  Flourens of course is famous for his empirical attack on phrenology.  His method involved ablation studies in a variety of animals--chickens, pigeons, frogs, dogs, rabbits--in which he successively removed larger and larger chunks of the cerebrum.  He reported in 1824 that “One can remove, from the front, or the back, or the top or the side, a certain portion of the cerebral lobes, without destroying their function. A small part of the lobe seems sufficient to exercise these functions.”  This led to his theory of the equipotentiality of the cerebrum, including the cortex, in contrast to the phrenological view, and provides a rational jumping off point for the 10% myth.  Kolb and Wishaw write, 

Perhaps the most commonly encountered Flourensian idea is in pedagogy, where it is expressed as the assertion that most people never use more than 10% of the brain. p. 9

A series of lectures in London by Charles-Édouard Brown-Séquard argued vehemently against the then-recent swell of empirical support for a localizationist view of cortex promoted by the work of Broca, Meynert, and Wernicke in the language domain and Ferrier, Bonchefontaine, and Fritsch & Hitzig in the motor domain. Citing the work of Flourens and reporting on new work of his own using a similar lesion based approach, Brown-Sequard argued that, 

Each half of the brain is a complete brain originally, and possesses the aptitude to be developed as a center for the two sides of the body in volitional movement as well as in all the other cerebral functions. Still very few people develop very much, and perhaps nobody quite fully, the powers of the two brains. [emphasis mine]

as regards localization of function, ... nerve cells endowed with the same function, instead of forming a cluster so as to be in the neighborhood of each other, are scattered in the brain, so that any part of that organ can be destroyed without cessation of their function.... all the symptoms of brain disease--such as paralysis, anesthesia, amaurosis, aphasia, insanity, convulsions, and the rest--are produced by the same mechanism, whether they arise from an irritation in any part of the trunk or limbs, or from an irritation in any part of the meninges or of the brain itself.

It's circumstantial, but the link between Flourens and the 10% myth seems plausible.

The lectures were published in The Lancet in 1876. Quotes are from the introductory lecture published in the July 15 issue. Scientific American published a commentary on the lectures providing the same quotes, thus possibly perpetuating the idea that few people fully develop the powers of the brain.