Thanks for the answer!

Hey there! I completed my senior project on James Joyce and Homer (and more specifically, the myth of blindness surrounding the two) this spring, so I can definitely help here. The problem is, I'm visiting family at the moment, so I don't have all of my sources in front of me. If need be, I'd be glad to try to find them online or clarify certain aspects of my response in a week or so when I'm back home, but for the time being I'm relying on memory, introductions to the Iliad and the Odyssey from copies that I do have with me, and quick Google searches to double check here-and-there.

Basically, "we don't know for certain, but possibly" is the most conclusive answer anyone could give you. Some scholars will argue confidently for his existence while others will vehemently argue against it, but--barring the invention of time machines--it seems unlikely that anyone will prove with 100% certainty whether a blind poet named Homer composed what we know of as the Iliad and the Odyssey or not. We do know, however, that most surviving commentaries from the Classical Era of Ancient Greece state pretty confidently that Homer existed and that he and Hesiod were greatest epic poets of a former age. There are even lost epics that were once attributed to Homer, some of them rather confidently (the Thebiad and Magrites, for example), and others that were of contested authorship even in Classical Greece (the Epigoni, among others). Plato praises Homer in the Republic, Aristotle wrote highly about Homer in his Poetics, and later Hellenistic writers, such as pseudo-Apollodorus, often included citations to the effect of "As Homer tells us..." when writing about the myths.

The so-called "Homeric Hymns" are an interesting focus as far as the "Homeric Question" goes, particularly because of the "Hymn to Delian Apollo". The speaker basically describes himself as "A blind man who lives in rocky Chios" whose songs will be remembered as the greatest forever. Other, later authors are now usually regarded as the authors of these texts, but the description given in the aforementioned Hymn would have been taken as a reference to Homer whether or not Homer was regarded as the actual author.

Another Hymn (I believe to Demeter) gives a reference to the Meles river, which might give another hint to Homer's life, or at least the stories later generations created about him. "Melesigenes" ("Μελησιγένης") was an epithet that meant "Son of the Meles" occasionally applied to Homer (Milton refers to him as such in Paradise Lost as well as Paradise Regained, for example). There are mythological or pseudo-biographical accounts of Homer's life called the Lives of Homer that often give Smyrna (valley of the river Meles) as his birthplace. The various Lives how the poet became known as "Homer" ("Ὅμηρος") which almost always refer to his blindness. I don't have my books in front of my, so this is a nice summary I found online:

Homer was given the name Melesigenes, Melesagoras or Melesianax at birth. He was born beside the river known as the Meles, which was near the Cymaean colony of Smyrna. The different variations of the name, Melesigenes, simply mean Meles-Born. Most researchers concluded that Homer was born beside the river Meles but there were others who insisted that he was the son of the river god, Meles.

An obvious question is: Why did a man named Melesigenes assume the name Homer? There are several explanations:

1. The Aeolians decided to abandon Smyrna as a colony and gave each citizen the choice of staying or leaving. Young Melesigenes said that he wanted to accompany [homerein] the leaders when they left the city and so he was thereafter called Homer. [Pseudo-Plutarch I, paragraph 3]

2. Although he was born with normal vision, Melesigenes became blind as a young man and was given the name Homer because it was a variation on the word for blind people, i.e. homeroi; [Pseudo-Plutarch I, paragraph 2] [Proklos Chrestomathy I, paragraph 3]

3. Another explanation for Melesigenes being given the name Homer was that he became blind as a young man and blind people required guides, i.e. homereuontes; [Pseudo-Herodotus, paragraph 13]

4. Finally, it was said that Melesigenes was given to the Chains [or a king of Persia] as a hostage and that the name Homer was a variation on the word for hostage, i.e. homeron. [Proklos Chrestomathy I, paragraph 3]

Now, as far as the actual writing process of these works is concerned, there are a few main theories. Since writing is virtually nonexistent in his surviving works, some take that as proof of Homer's own illiteracy, suggesting that he was possibly a master poet of the oral tradition whose works were later written down and/or compiled by others. After Robin Wood's Essay on the Original Genius of Homer in 1769, it became common for scholars to try and pick apart the Iliad and the Odyssey into smaller poems that had been pieced together over the years, but this has widely been debunked, partially because no two scholars could agree where to draw the lines, and these separations often made little sense to anyone other than the person who made them. Similarly, historians have tried to take apart sections based on what may seem to be later interpolations (for example, bronze spears mentioned in one place whereas soldiers have iron in another), but this proved to be largely ineffective. His works have been hard to pick apart linguistically as well, considering that "Homeric Greek" is an Ionic (Asia Minor) dialect that tends to draw vocab and/or grammar from other dialects and regions. Also, Homer's works were more or less edited to remove interpolations or inconsistencies in the 6th Century BC by the Athenian tyrant Peisistratos, so it would be impossible to tell today exactly which words came directly from Homer and which may have been changed before the more-or-less canonical versions that have been passed down through the centuries.

For the more modern approach to the "Homeric Question," Milman Parry did a lot of work researching Homer's meter that was later incorporated and furthered into Albert Lord's research about illiterate bards in the mid 20th century who composed epics of incredible lengths, debunking the "no one could memorize a poem that long" theory that has been used off-and-on to discount the notion of a singular poet of the Iliad and the Odyssey. A wildly over-simplified gist of their work is that the epithets and repeated sections of Homer's poetry are proof of an oral composer. These memorized sections could be used to fill out a line to fit the dactylic hexameter (hence why in the Iliad you may have "the long-haired Achaeans" in one line and something like "the mighty Achaeans" in another), or, in the case of longer repeated sections, be used as a type of space-filler by the poet in order to plan ahead for more nuanced sections of the poem that required more thought. Based off their work, many scholars today would agree that one poet from the epic oral traditiong who composed his songs at the fringe of the "Dark Ages" of Ancient Greece (named so because we know almost nothing about them) and the invention of their writing system composed the Iliad and the Odyssey, which may have been dictated to scribes, or he may have learned the writing system and wrote them himself. Obviously there are people who still disagree with that, but it seems to be the leading theory today, and nothing I came across while doing my research led me to believe otherwise.

Some of this is a mess, but I'll stop for now before I spend an entire day writing about this. If you have any further questions or would like some clarification, let me know!