The word αἰδώς means awe, shame, or respect. There are related words such as αἰδοῖος. I feel like I ought to be training my brain to recognize inflections in order to get clues as to meaning, but as far as I know, -ώς is used to form adverbs. Is there some way to understand why it's used in this word? Is it a tag for an abstraction? For that-which-is-caused-by? For forming a noun out of an adjective? Is this just a phonetic thing in this word, rather than a semantic one?
"Is this just a phonetic thing in this word, rather than a semantic one?" Yep. In fact, as Smyth says, αἰδώς is the only such "-οσ- stem" word in Attic. (In Homer you will also find ἠώς "dawn", which in Attic declines as an "Attic declension" noun, on which see below). So it might as well be considered irregular. The full declension (it only occurs in the singular) is Nom. αἰδώς, Gen. αἰδοῦς, Dat. αἰδοῖ, Acc. αἰδῶ, Voc. αἰδώς. The oblique case forms result from contraction: e.g. historically the genitive was αἰδόσ-ος, but the intervocalic sigma was regularly lost bringing the two omicrons into contact, αἰδόσος > αἰδόος > αἰδοῦς, and likewise in the dative and accusative.
There is, however, another slightly more populous set of nouns in -ως, known as the "Attic declension", which are basically an odd subclass of the second declension; Smyth discusses these here.
Note that the accent of αἰδώς rules out adverbhood, since when -ως adverbs are accented on the ultima they are always circumflexed, -ῶς.
@TKR is right about the specific case of αἰδώς and mentioned the Attic declension, but there's more to say: there is a good number of Greek nouns ending in -ως even outside the Attic declension, and the question of whether it's a meaningful suffix for forming words out of other words there is worth asking.
Going through Wiktionary's list of Greek nouns and filtering out all of the ones that aren't Attic or are transparently perfect participles used as nouns (that means you, ξυνειδώς), we can detect a few basic groups:
Your lonely -οσ- stem αἰδώς.
Aforementioned nouns in the Attic declension that are the result of quantitative metathesis: λεώς (< *ληός, cf. Epic, Doric, and Koine λᾱός) and the name Ἡγησίλεως (Koine Ᾱ̓γησίλᾱος), νεώς (< *νηός, cf. Doric νᾱός), περίνεως (cf. Doric νᾱός but also Attic ναῦς with an υ, which suggests this compound was a loan), φλέως (< *φλήος).
Words with a stem ending in -ωτ- and an apparent associated verb: γέλως (γελάω), ἔρως (ἐράω), φῶς (φάω; φάος also attested, stem φάε- (!)), possibly χρώς (χρίω), possibly Εἵλως (ἁλίσκομαι). There may be a connection with the active perfect participle in some way, but as far as I know the details of these formations (if they are all the same kind of formation) haven't been worked out. Either way, this is not a productive category: it's not possible to form e.g. ˣτίμως out of τίμαω.
Short words with a stem in -ω-: δμώς, ἥρως, θώς. The fact that the first of these reflects the zero-grade of the Proto-Indo-European root also seen in δάμνημι and the second probably reflects a root not otherwise seen in Greek (other than perhaps the name Ἥρα) but seen in Latin servo confirms that these are very old (θώς may be related to θῶμαι). The -ω- here certainly represents some PIE suffix, but it's not one that's productive in Greek anymore.
Three very old kinship terms, the first of which is probably unrelated to the other two: γάλως (Aeolic γαλόως; cf. Latin glōs), μήτρως, πάτρως (cf. Latin patruus). All three of these are in the Attic declension, apparently, but if they're the result of quantitative metathesis it's not very obvious how. They can also be declined according to the third declension, in which case the first has a stem in -ωτ- and the other two in -ω-, but those can't blindly be taken as etymological (Latin glōs, glōris suggests an original s-stem for γάλως; patruus is thematic, which can't be original either).
Animals in the Attic declension: λαγώς 'hare; sea hare', ὀρφώς 'dusky grouper' (a kind of fish apparently), ἀχαρνώς '(probably) bass' (also fish), ταώς 'peacock'. The first one is from an adjective *λαγοωυσός 'floppy-eared', so the -ω- in λαγώς represents a series of contractions. The fish names might be analogical? The etymology of ταώς isn't known.
(Third-declension dental stem ῥινόκερως, for the record, is a post-Classical coinage, as you can tell by the accent, and it's originally an adjective.)
Miscellany: dental stems εὐρώς, ἱδρώς, φώς (may belong with the third bullet point); Attic declensions of unknown etymology ἅλως (may be related to ἀλέω), ἀρηβώς (may have an analogical -ς; ἀρηβώ also attested); and the aforementioned Attic declension/very unique ἕως.
So the first observation here is that there isn't one suffix -ως: there are at least three (Attic declension, stems in -ω-, stems in -ωτ-), and each of those in turn represents the merger/levelling of several different things. More importantly, though, all of these are very small groups and none of these formations is predictable—these are not productive suffixes.
If you see a word in the Attic declension ending in -εως you can predict that most other dialects will probably have -ᾱος there, but apart from that there is no real conclusion you can draw from a noun ending in -ως.