What can be learned about the character of Penelope in her interview with the "beggar"?The character of Odysseus in his interview with his wife?
There are quite a few things that we can take from this interview.
First and foremost, it obviously suggests that she is not over the loss of her husband, and that she has an interest in him returning. Of course, her continued unweaving of the death shroud shows this, too, but even when faced with the fact that she may have to choose a suitor, she's still inquisitive.
Additionally, it shows she's smart. She asks questions, and wants details, and this allows Penelope to find out more and more about not just her husband, but also the stranger...who are, of course, the same person.