Here's some information culled from Gardner Read's classic text Music Notation: A Manual of Modern Practice (1969), which begins with an overview of the history of Western musical notation:

Nearly 3000 years ago, Greeks employed at least one system of letter notation, letters corresponding to scale notes (including octave distinctions.) In the Pre-Christian era, there were four such systems (that we know of) in use. Changing the orientation of the letters indicated raising or lowering the notes, as with our sharps and flats. Duration was sometimes indicated by signs used in conjunction with the notes.

Around the middle of the 4th c. CE these systems contained over 1600 different signs, symbols and letter forms. They used one system (based on Ionian letters) for vocal music, another (based on Phoenecian letters) for instrumental music.

A 6th c. Italian monk, Boethius devised a similar system using Latin letters, but his system did not gain any widespread use.

The neumes of Gregorian chant first came into being in the 6th century. They were based on the acutus and gravis of Greek prosody, indicating rising and falling inflection. Initially, only relative, rather than precise, pitches were notated, with no durations indicated.

A (six-lined) staff appeared in a 9th c. theoretical tract, Musica Enchiriadis, which also gave examples of early polyphony, mostly in parallel organum. However, the system used words placed between the staff lines, rather than neumes, to indicate pitch.

Fixed pitch using the neume system and a single staff line (F) appeared in the 10th c. The various neumes were placed at their relative pitch distances above and below the staff line. Soon after, a second line (C) was added above, to allow more precise placement of the higher notes.

More lines were added, as demanded by the music, until Guido d'Arezzo standardized and popularized a four line staff in the 12th c. While we owe him a great debt for this, his only innovation was to introduce the scale syllables (ut re me fa sol...), after noticing that six successive lines of a Sapphic hymn to St. John the Baptist began on each successive note of the scale. He named the pitches according to these first syllables, which made it much easier for him to teach chants to even young students. Read doesn't describe how d'Arezzo handled the seventh scale pitch, which was not named si (ti in Germany) until several centuries later. He does say, however, that si was supposedly derived from the initial letters of Sancte Iohannes (St. John).

The flat and natural sign (a rounded and squared b respectively) arose about the 11th c., with the sharp and other accidentals appearing later still.

The problem of notating durations and rests, as required for coordinating more sophisticated polyphony, did not begin to be adequately addressed in a consistent way until the 13th c.

I have read elsewhere that the Egyptians did indeed have a system of musical notation that predated even the Greek systems, but few details of it were given in that source (which I no longer recall.)

Cheers!