The question is talking the dominant current view of "Homer", in that this was a person who created these poems sometime around the 8th or perhaps 7th centuries, and then someone familiar with them sat down with a scribe to transcribe them into written Geek sometime around the 8th Century BC. If one accepts that view, then both Homer's literate contemporaries and the scribe(s) in question would have been using an early version of The Greek Alphabet.
The Greek Alphabet was the worlds first true alphabet. It was developed from the Phoenician script from the 9th to 8th centuries BC. This Phoenician Alphabet actually had no vowel glyphs, since Semitic languages don't need them so much (so it was arguably the true innovation here). Either way, it was the new Greek Alphabet, with its easier-to-learn glyph set of around 20, that brought the Greeks back into history as a literate culture once more.
These Greek Alphabets weren't really standardized until the 4th Century BC, after the Peloponnesian War. I don't think we have any real clues about which part of the Greek World Homer hailed from, and in any event the whole area would have been speaking mutually-intelligable dialects of Greek at this point, so an illiterate bard probably would neither know nor care, so I don't think we have a way to pick one of the variants. However, here's a chart showing their names and how similar they were.
I'll also post here a map I found of the places mentioned in the Iliad. Perhaps someone can glean out more clues to the likely home of Homer from that, but all I'm pretty sure about is that he's probably from the Peloponesian half (where nearly all the heroes and most of the mentioned places are from), rather than the Asia Minor half. So that may argue against the Ionic variant.
However, Ionic was very influential, as shown in the fact that its script eventually became the basis for the new standard in the 4th Century. Also, linguistic analysis of the dialect used in both works shows it to be largely Ionic. I don't think we can say that this means Homer himself used that as his native dialect, but it does seem to indicate that the most influential scribes who wrote down his stories did. However, the poems themselves still contain a lot of artifacts that are useful to bards reciting them from memory, which I'd say in fact argues for Homer speaking an Ionic dialect, either natively or as his professional dialect for the purposes of composition. Either the Ionic or the Euboan variants of the script would be the natural literary expressions of that dialect.
As a bonus answer, I'd say if this were a question about literate people from the time and place the Iliad and Odyssey are set, most likely a such a person would have been using the script we know today as Linear B, or something closely related to it.