[1] https://en.wikipedia.org/wiki/Homeric_Question
[2] https://en.wikisource.org/wiki/The_Authoress_of_the_Odyssey
The Iliad and the Odyssey use very similar artificial poetic languages and vocabularies (though some words appear for the first time in the Odyssey and most of them are words that are expected to be more recent words).
Even so there is a very noticeable difference in style between the two works, and the easiest way to describe this difference is to say that the Iliad seems masculine, while the Odyssey seems feminine, i.e. the former is like an action movie, which spends a lot of time with the description of matters interesting for males, e.g. about efficient ways of killing or maiming your opponents or of gaining glory on the battlefield, while the latter is like a chick flick, where the main interests are about love and romance, stories about powerful independent women, descriptions of various female skills, clothes, food and gardens, and it includes even feminist complaints about the lack of equality between sexes.
It is very unlikely that we will ever know anything certain about the identity of the authors of the Homeric poems, but reading carefully the two texts, especially in original, gives the appearance of two closely related authors, but nevertheless of different sex.
Devil's advocacy:
Even if the Odyssey is arguably a relatively paler reflection of the Iliad in terms of mythological weight across the western corpus (ie: centrally important myths reflected within other myths), while still being a monster in its own right, it would be a monumental lifetime feat for one woman to acquire the deepest mythological and even religious (apocalyptic) knowledge it would have taken to write the Odyssey. It's still an incalculably skilled work.
In all, I'd lean against the one woman theory. But it wouldn't surprise me either. Authors and artists often had advisors on classical subject matter that would have been mostly mastered by those with expensive educations.
Homeric Question and historicity:
No one who is a serious student of mythology thinks that there is a real controversy over whether or not works such as the Iliad fall into dichotomous categories of true or false. What these myths are meant to describe are archetypal repeating events. That is, they are both true and myth. As is the case for most long persevering myths. No one should allow a little bit of allegory to fool them.
Butler's theory is nice and all, but I would give significantly more weight to what ancient writers had to say about Homer.
On the contrary, most of its stories relate to extremely tales and folklores - ie Polyphemus.
Have fun : https://www.scientificamerican.com/article/scientists-trace-...