Perhaps the most important value that is illustrated in this scene is that of "xenia" or hospitality towards strangers. This was not just a cultural value but a religious one. Zeus was the "god of strangers" and to mistreat a stranger was to call down the wrath of Zeus. Penelope's noble and devout character are shown in the way that she offers hospitality even to a ragged stranger and uses her limited power to enforce this upon both her servants and the suitors.
Odysseus' discussion of his past sufferings also shows the degree to which the Greeks admired the ability of a person to endure hardship.
Both Penelope and Eurycleia demonstrate the female virtues of loyalty and domesticity. While the culture is shown as very hierarchical, and slavery taken for granted, domestic (as opposed to field) slaves are considered part of the household. We also see in the way that Penelope is surrounded by female servants a homosocial tendency in Greek household organization, in which women, whether free or slave, tended to associate mainly with other women.
Her words show that the Greeks were not foolish or naïve, for she asks for proof that Odysseus (in disguise) really met her husband. However, when he gives it to her, she weeps, and offers him considerable hospitality:
“Stranger, I was already disposed to pity you, but henceforth you shall be honored and made welcome in my house. It was I who gave Odysseus the clothes you speak of. I took them out of the store room and folded them up myself, and I gave him also the gold brooch to wear as an ornament. Alas! I shall never welcome him home again. It was by an ill fate that he ever set out for that detested city whose very name I cannot bring myself even to mention.”
This accents the Greek value of xenia, the Greek value of guest/host obligation, which we often translate as hospitality. The Greeks valued it highly. (Notice that Penelope also weeps; passion was valued, as was pity.)