Q:

How did the theory of the Horus-eye fractions originate?

In a book, I saw a short anecdote about something called the "Horus-eye" hieroglyph, supposedly used in parts to represent fractions (the book dates to 1997). Wikipedia has an article on the symbol and states

Studies from the 1970s to this day in Egyptian mathematics have clearly shown this theory was fallacious and Jim Ritter definitely showed it to be false in 2003. The evolution of the symbols used in mathematics, although similar to the different parts of the Eye of Horus, is now known to be distinct.

I detect a slight lilt towards a bias against the symbol there, given that I've read other (more recent than the book) sources that either omit the "studies from the 1970s" and Ritter's work or barely mention it altogether - not that I'm doubting it; I'm only noting that perhaps there isn't as much certainty as Wikipedia claims.
If we use the hypothesis that the symbol did not represent fractions then someone in the modern era must have come up with the theory. Who was it? Wikipedia alludes to Gardiner's sign list, but I have no idea if the hieroglyph is represented there merely in an encyclopedic sense, e.g. Gardiner catalogued it from someone else.
Here, by the way, is a diagrammed representation of the symbol:

Wikimedia Commons

A:

Ritter himself gives a detailed historical account in Closing the Eye of Horus: The Rise and Fall of "Horus-eye Fractions". Egyptians had two forms of writing, hieratic (cursive) and hieroglyphic (pictographic). Ritter traces the origin of the myth to "“central dogma” of our ideas about Egyptan writing: all hieratic signs go back ultimately to pictographic signs. The idea of organizing hieratic signlists by hieroglyph is one expression of this and, as we have seen, it was precisely in just such a context that the search for the “hieroglyphic originals” of the capacity metrological signs led to the Horus-eye hypothesis". But analysis of documents shows that in Egypt, as in Mesopotamia, numerical and non-numerical signs evolved quite differently. Ritter's view remains the consensus, see e.g. The Mathematics of Egypt, Mesopotamia, China, India, and Islam edited by Katz.
The story begins with Möller's 1911 Hieratische Paläographie ("CSM signs" below are what Ritter calls the "fraction" symbols):
"Möller presents essentially two arguments for the identification of the CSM signs with parts of the completed right eye of Horus: 1° the proximity of some hieratic and some hieroglyphic forms of the signs on two unpublished votive cubits and 2° the visual similarities of the remaining signs, which “can be traced back to parts of the eye”". Peet already called this idea into question in 1923, because "1° the hieratic forms “show but slight resemblance to the parts of the eye” and 2° the hieroglyphic forms are much later, they do not predate Dynasty 18 or, for some forms, Dynasty 20. The frequent use of the hieratic forms in the medical texts leads Peet to speculate that the Horus-eye identification was “an ingenious discovery of some [medical] scribe” since the sacred eye was “so closely connected in Egyptian eyes with healing power.”"
Nonetheless, Gardiner accepted Möller's "strong thesis", as Ritter calls it, in Egyptian Grammar (1927), even reflecting the eye from right to left to make the signs fit. After the publication of Moscow papyrus in 1930 Neugebauer objected to that, and pointed out that there was little resemblance in 1/16 and 1/32, arriving at Peet's "weak thesis" that identification was of "late date".
Still, "...by the beginning of the nineteen-thirties, the Horus-eye interpretation had entered the two canonical grammars and, especially in the form given to it by Gardiner, was to become the standard interpretation for Egyptologists and historians of science in the succeeding decades. This was true very early; indeed neither Struve nor Neugebauer nor Vogel cites Möller’s article but only Gardiner".
Things changed after publication of new documents in 1960-1970s, including those unpublished by Möller, and a new look in their light at the old documents. Ritter gives a detailed discussion, summarized as follows:
"The trend of the evidence of third-millennium sign forms that have accumulated since the end of the debate in the nineteen-thirties is clear. The further back we go the further the hieratic signs diverge from their supposed Horus-eye counterparts. The inconsistency between the hieratic signs for “1/16” and “1/32” and their purported hieroglyphic originals, already signaled by Neugebauer..., is even greater for the original third-millennium forms and the others reveal similar difficulties. The hieroglyphic “1/2” signs vary widely among themselves and all differ from their New Kingdom descendants".
So much for the strong thesis. And the weak thesis? 
"If it, unlike the strong thesis, cannot be ruled out completely, the positive evidence for it seems to be meagre in the extreme... The beginning of the replacement of the special CSM signs by their general fractional equivalents is to be located just at this period when, in the weak thesis, the idea that the classical CSM signs form an important conceptual and religious unity begins its career. The votive cubits that are supposed to enshrine this idea have proven to be much more ambiguous and unclear than foreseen... But even if the cubits did represent an attempt to reinterpret the CSM signs in terms of Horus-eye parts, a speculation in the milieu which produced votive cubits does not automatically mean that “the Egyptians” thought like that; for example, those Egyptians whose task it was to engrave hieroglyphic inscriptions on temple walls. Theological or any other constructs of one community do not necessarily propagate to every other...".

