Wordplay is pretty common in early-to-classical Greek poetry. But ... don't think of it as puns.

When Odysseus tricks the Cyclops with the name 'Nobody' (Outis) and calls that a metis -- μῆτις 'cunning trick' ~ μή τις 'nobody' -- it isn't a joke. Or, it isn't JUST a joke.

It's be more on the mark to think of it as word magic. It's like when the Derveni papyrus equates 'being beneficial' and 'prophesying', because χρῆσαι can mean both. Or when it relates Kronos/Chronos to κρούω ('collide'). There's something mystical about it.

A famous one is in Aeschylus' Agamemnon. At one point the chorus sings about evil Helen, and calls her

'ship-destroyer, man-destroyer, city-destroyer'
ἑλένας ἕλανδρος ἑλέπτολις
helenas helandros heleptolis
(Ag. 689)

This definitely isn't a comedy by the way.

But the biggest cluster of wordplays, I think, is around Odysseus. Odysseus' name is a big deal in the Odyssey. He's anonymous in the proem (until line 21); he's anonymous to the Phaiakians for 3 books; he's literally Nobody to the Cyclops; he doesn't name himself to his son.

In fact when he does reveal himself to Telemachus, he says:

'I am no god...
but I am your father'

οὔ τίς τοι θεός εἰμι...
ἀλλὰ πατὴρ τεός εἰμι

ou tis toi tʰeos eimi...
alla pater teos eimi.

(Remember Outis = 'Nobody': 'I am no god' ~ 'I am your Nobody'.)

The wordplay famously becomes part of the story in book 19: he was called Odyss-eus because his grandfather 'angered' (odyss-amenos) many people.

More striking, but obscured in translation: Athena's speech to Zeus in book 1 plays on his name with dys-, odys-, and de-/da- sounds.

Parry didn't call this 'wordplay' -- he used the French term 'calembour' instead. I guess that has more literary dignitas.

Further reading: Ahl, in Arethusa 35 (2002); Louden, TAPA 125 (1995); Olson, ICS 17 (1992); Austin, CSCA 5 (1972).