Greek 'without a name' — related to 'synonym,' 'antonym,' and even 'noun,' all from PIE 'name.'
Of a person: not identified by name, of unknown identity. Of a work: of unknown authorship.
From Late Latin anōnymus, from Greek anōnymos (ἀνώνυμος, nameless, without a name), from an- (ἀν-, the privative prefix, without, not) + ónyma (ὄνυμα), the Aeolic dialect form of ónoma (ὄνομα, name), from PIE *h₁nómn̥ (name). The PIE root *h₁nómn̥ is reconstructed as the original proto-word for name, descending into Latin as nōmen, Old English as nama, Sanskrit as nāman, and Greek as ónoma. The privative an- (before vowels) and a- (before consonants) is itself from PIE *n̥-, the zero-grade of the negative particle. Anonymous entered English in the early 17th century. The same root gives synonym (same-name), antonym (opposite-name), homonym
The PIE root *h₁nómn̥ (name) is one of the best-attested words in reconstructed proto-language because it is preserved in nearly every branch: English 'name,' Latin 'nōmen' (giving 'noun,' 'nominal,' 'nominate'), Greek 'onoma' (giving '-onym' words), Sanskrit 'nāman,' and even Old Irish 'ainm.' The word 'noun' literally means 'name' — a noun is a naming word. And 'anonymous' means 'un-named' — the negative prefix flipped onto one of humanity's oldest words.