The Greek word εὐγενής ‘noble’ comes from εὖ ‘good’ + γεν- ‘breed’, but where does -ής come from?

-ής, -ές is a suffix used to form adjectives. The entry on -ής, -ές in Smyth’s Grammar §858, reads (Herbert Weir Smyth, A Greek Grammar for Colleges):

5. ες (nom. –ής, –ές): primitive: ψευδ-ής false (ψεύδ-ω deceive), σαφ-ής clear, πρην-ής prone, ὑγι-ής healthy. Very common in compounds, as ἀ-σφαλ-ής unharmed, secure (ἀ-priv. + σφαλ- in σφάλλω trip).

So the compound adjective εὐ-γεν-ής ‘good + breed’-ής for ‘noble’ is analogous to ἁ-σφαλ-ής ‘un-tripping’-ής for ‘secureʼ.