In antiquity, every city had its own variant of the Greek alphabet; they varied not only on shape of letter, but also on which letters they used.
Athens undertook a spelling reform in 403 BC, under the archonship of Eucleides, which adopted the Milesian variant of the Ionian alphabet, including the letters eta and omega. (That alphabet was already widely used in Athens, but it was not the original local variant.)
The Modern Greek alphabet is the Euclidean Athenian alphabet, because the orthography of Modern Greek is historical. There are letters in the Euclidean Athenian alphabet that are redundant in Modern Greek (η υ ω); but it’s only in the USSR of the 30s that any serious attempt was made to write Modern Greek phonetically.
Of the archaic letters, pre 403 BC, only one truly matters: Digamma, ϝ. It was the Ancient Greek /w/. The sound does show up in papyri of Sappho and Alcman. We can also reconstruct it in Homer, though it is missing in our text of Homer. The text we have of Homer was established in 6th century BC Athens; and the sound (and thence the letter) was long extinct in Athens by then.
Of the others: San (letter), ϻ, and Koppa (letter), ϙ, were adopted because Samekh and Qoph are letters of the Phoenician alphabet.
Greeks originally wrote <q> before back vowels and <k> before other vowels; so they wrote Ϙόρινθος <Qorinthos> for Corinth, and in fact the koppa became a symbol of Corinth. Eventually, they worked out that those were just allophones, and they dropped the koppa entirely except for counting (it’s the number 90).
Shin (letter) and Samekh were differentiated in Phoenician, because Phoenician had both /ʃ/ and /s/; Ancient Greek did not, and once people stopped robotically copying the Phoenician alphabet, each town picked one or the other of sigma or san to write /s/ with; no town kept both.
Modern publications of any literary texts that survived via the papyrus tradition went via Athens and its 403 BC spelling reform. Digamma is rare; san and koppa are non-existent, even though Aristophanes refers to san-branded horses. (Just as Qorinth used the koppa as its emblem, Sicyon used the san.) You’ll only ever see them in publication of inscriptions that did use them; and even then you won’t see them often.
In the process of establishing the Unicode repertoire for Greek, I came across a few more archaic letters. In fact, I came up with the name of one: Tsan, the Arcadian variant of San, which apparently was used for /ts/. The variant letters that were phonetically distinct, and which will show up in publication of inscriptions, were Heta, the eta-equivalent used to mean /h/ and not /ɛː/; the Sampi, an Ionic innovation which might have been pronounced /ʃ/ or /sː/; and the Pamphylian digamma, which looked identical to tsan but was likely pronounced /v/, as opposed to the normal digamma’s /w/.
One variant that didn’t make it in the repertoire was the Corinthian single letter for <ei> /eː/. It didn’t make it into Unicode, because the letter for it was in fact <E> (Corinth had a different letterform for epsilon.) No epigraphist has published it as anything but ει or, if they’re being extra careful, <Ε> in a different font. (Google Dweinias for the most important inscription.) Ditto the corresponding Boeotian letter, which looks identical to the tack-shaped heta, <Ͱ>. (http://www.opoudjis.net/unicode/…)
I have been responsible for conflating Pamphylian sampi and normal sampi in Unicode, even though they in fact look different, and I have posted my apology for that in Nick Nicholas’ answer to If you were allowed to add a symbol to unicode, what symbol would it be, and what would it mean?
Oh. Sho (letter) is Bactrian for /ʃ/; Bactrian used the Greek alphabet, but I don’t think sho counts here.