I don't happen to think she does, but I'm interested in what anyone else has to say about this.
Her son often accuses her of enjoying their presence -- never choosing one, but never wanting them to leave. But I think this is just bitterness on his part. Especially since, from things he himself *and* Penelope have said, his own feelings about his mother's possible remarriage may be mixed.
On the one hand, he wants her to be faithful to Odysseus. Like a child, even if his father is dead, he never wants his mother to have another husband.
But as a man, coming into a man's estate, he wants his mother to marry another wealthy man, so that Telemachus can take over Ithaca -- and so there'll be anything left to take over. The suitors, as he often complains, are draining the place dry.
I think of Penelope as very much a victim of the suitors abuse of Greek hospitality customs.
But her dream about the geese and the eagle might suggest that, as her son once said, her vanity is fed by the suitors' constant presence.
I think it comes down to Greek custom. Since Telemachus is of age, if his mother were to remarry, first she would have to be sent back to her parents home and the "bride price" would have to be repaid, probably by Telemachus. Then she would have the opportunity to remarry. Wouldn't THAT be a hoot! How old must she be? At least 35, back in her father's home? Telemachus may not want his mother to remarry because he can't afford it!
As we were talking about last night (woohoo!), there's still some serious doubt among scholars as to who would pay a "bride price" for someone in Penelope's position.
Just going by textual evidence, the money and goods do seem to be flowing both ways. Penelope says that, if she has to remarry, she must choose the man who gives her the richest gift. So the guys are obviously having to pony up in order to prove their eligibility.
But it's also mentioned a few times in the poem that Penelope would marry from her father's house. In fact, though Telemachus was ambivalent at best about his mother remarrying, one thing he came down firmly about was that he wasn't going to send his mother away from Ithaca -- send her back to her parent's home, where she could be properly married. But the language Athena uses about paying the bride price for her is ambiguous, if I'm remembering right.
It's possible that bride price customs differed by region, or shifted over time. This happened in India. Originally, a man who wanted to marry a woman there would offer money or goods for her, to show that she was valued and to make up for taking her away from her parents. Later, it shifted to the British custom of the woman having to bring a dowry to her marriage. This led to a marked increase in female infanticide in poor families, who couldn't afford a dowry for daughters, and in the comparatively recent spate of "bride burnings," where women who haven't brought enough money to their arranged marriages are assaulted and even murdered by their in-laws (who then claim that it was suicide on the bride's part).
