Writing a poem down originally had nothing to do with composing it and even less with 'writing a book'. In archaic Greece, writing served one main purpose: preserving for posterity. In other cultures, including the Mycenaean, administration was a major factor in the use and even invention of writing, in Greece there are no traces of that. The word sēma which Homer uses for written messages (Il 6.168) means "sign" with many applications but one of them is grave, tomb, grave-marker (Il 7.86-, 21.322, 23.255 and the "sēma" of a man long dead that serves as the turning point in the chariot race (Il 23.326-). The allusion is clear: the Iliad itself is a sēma, a grave marker for Patroclus, the best of the Achaeans (and of course for the poet himself). It must have been an exciting discovery for a poet: a way of preserving your work for posterity, where normally it would die with the singer. There may be apprentices, a younger generation, but they will sing their own songs, not yours. This brings us to a related function that a written text may have: it can be used for teaching.
There is some writing-related circumstantial evidence:
Admittedly this is meagre evidence, but it does make sense (see below) and it fits in with what we know about the Homerids and about the history of writing. It also offers an explanation for the remarkable degree of preservation of the text.
If you are a traditional professional singer you may have an apprentice who follows you around from performance to
performance, carrying your lyre for you and trying to pick up the technique of epic singing by listening to many
performances. This is not a very efficient method of learning and you probably cannot have more than one or two such
apprentices in a lifetime. They will know the "formulas", the storylines and the technique of performing them before
an audience but they will not know your composition by heart. The idea that a person could learn it by heart by just listening to performances seems rather farfetched to me. So, what to do if you are a not-so-traditional
world-famous singer who composed a monumental work, setting afire the people's enthusiasm wherever you go, inspiring
young men to come to seek you out because they want to learn this?
You start a school. You find a place out of the way of population centers but easily reachable and there you receive them and teach them your songs. For this you need a written copy. Without that even a voice of bronze will not be strong enough to keep singing the songs to them. So first you will have to teach them to read and write using your own poems. This works both ways, for your pupils can then help you with the major task of writing down the complete poems. Several problems have to be solved: what to write on (cow- and sheepskin? see Hdt. 5.58, possibly also Il 17.389-93), what to write with (a carefully shaped pen of olive wood, its point hardened in the fire? Od 9.319-) and what to use for ink.
All this may give you the reputation of being the world's first schoolmaster and, since they do not know your real name, the nickname "Phoenix" because of the Phoenician letters you are always occupied with.
Where would you live? You will want to be out of the way of powerful noblemen and cities who would use your fame for
their own purposes. Therefore mainland Ionia and mainland Greece are not attractive, an island seems most suitable -
somewhere in between those who live where the sun rises and those who live where it goes down. Since you are always
anxious to hear news from your old hometown Smyrna, you will want to be near one of the main shipping routes.
Now there happens to be a fertile and wooded island stretched out before the Cyclops' harbour, not near nor very far...there are soft and well-watered meadows and a harbour where you can beach your vessel and stay until the wind becomes favourable...(Od 9.116-42). There is a place where those sailing the route north of Chios across the Aegean to Euboia (Od 3.170-) actually do that(1).
Having finished the writing down of your magnum opus, the Iliad, and being too old to go on extensive travels like you used to, you spend your time composing a second large poem, meant to be performed by your pupils. Since none of them is able to string your lyre, they learn to beat the rhythm with a wooden staff. So they go to the festivals like Delos and to funerals and win the tripods in the singing competitions.
In this way you could live out your life. Your daughter Eidothea will take care of you while you receive visitors and cultivate your position as an immortal who knows the sea in all its depths and who can magically change into every phenomenon on earth or in heaven. You can even tell those visitors which of the gods it is that prevents them from reaching their home. And, of course, you will be able to solve every riddle that they put to you.