Strabo, who was in Egypt from 24 to 19 BC and gave us detailed accounts of the Museum, claims the inspiration for the Library's organization - at least initially - was Aristotle (emphasis mine):
From Scepsis came the Socratic philosophers Erastus and Coriscus and Neleus the son of Coriscus, this last a man who not only was a pupil of Aristotle and Theophrastus, but also inherited the library of Theophrastus, which included that of Aristotle. At any rate, Aristotle bequeathed his own library to Theophrastus, to whom he also left his school; and he is the first man, so far as I know, to have collected books and to have taught the kings in Egypt how to arrange a library.
Source: Strab. 13.1.54.
The story seems likely; Aristotle was the father of taxonomy and his personal library was one of the earliest acquisitions of the Library. The early librarians probably modeled their own approaches upon Aristotle's. However, we don't have any details on how Aristotle organized his library, and to what degree the early librarians adopted his methods.
What we do know, is that Zenodotus of Ephesus, the first librarian according to the Suda, arranged books in different rooms according to their subject matter. Then, within each room, he arranged the books alphabetically (by their author's name). His system may seem rudimentary by today's standards, but for the time it was revolutionary. In fact, it's the earliest known instance of alphabetic organization.
Callimachus of Cyrene furthered Zenodotus system in his magnum opus, the Tables of people who distinguished themselves in all branches of learning. Pinakes, as the work is better known, is a bio-bibliographical survey of the more significant Greek writings, divided in 120 books. According to Wikipedia:
Callimachus' system divided works into six genres and five sections of prose. These were rhetoric, law, epic, tragedy, comedy, lyric poetry, history, medicine, mathematics, natural science and miscellanies. Each category was alphabetized by author.
Unfortunately only a few fragments of the Pinakes survive, but from those fragments we know they included biographical data, including the author's other works, and general notes for each work (e.g. its extend). Callimachus' Pinakes were the key to the vast collection of Library, and were constantly updated by subsequent librarians and grammarians, as the collection grew.
1 Πίνακες των εν πάση παιδεία διαλαμψάντων.