Epithets add a bit of color and also fill out the meter when the name on its own doesn't quite fit. In addition, epithets serve as a mnemonic device reminding listeners that they have, indeed, already heard mention of the character. The epithets, generally compound adjectives, are picturesque, which certainly helps make the assignment of character to epithet memorable.
Most of the important people in the Iliad have a special epithet that serves as an extra name. Athena is the only one described as glaucopis 'grey-eyed'. She is called thea glaukopis Athene 'goddess grey-eyed Athena' and also Pallas Athene 'Pallas Athena'. On the other hand, Hera shares her epithet leukolenos 'white-armed'. Hera does not however, share the longer epithet thea leukolenos Hera 'goddess white-armed Hera'; nor does she share the epithet bouopis potnia Hera 'cow-eyed mistress/queen Hera'.
Homer never calls the Greeks 'Greeks'. Sometimes they are Achaeans. As Achaeans they receive the epithets 'well-greaved' or 'brazen-clad Achaeans'. The title anax andron 'lord of men' is most often given to the leader of the greek forces, Agamemnon, although it is also given to others. Achilles receives epithets based on the swiftness of his feet. Odysseus is polutlos 'much-suffering' and polumytis 'of many devices, crafty'. There are other epithets for Odysseus beginning with polu- 'many/much' that Homer selects on the basis of how many syllables he needs for the meter. The messenger goddess, Iris (note: the messenger deity is not Hermes in the Iliad), is called podenemos 'wind-swift'. Perhaps the most familiar epithet is the one used for the passage of time, rhododaktulos Eos 'rosy-fingered Dawn'.