In The Odyssey, why do Telemachus and his mother need Odysseus?
Unlike modern times, where a mother and child the respective ages of Penelope and Telemachus can get along pretty darn well without the "man of the house" being around, in the culture depicted in Homer's Odyssey, this does not seem to be the case.
In modern times, Penelope and Telemachus could call the police to help protect them from unwanted intruders. Unfortunately, the culture of Homer's Odyssey placed a much heavier burden on the host, who was expected not to injure or violate a guest, unless that guest was behaving in a very inappropriate manner. Even though Telemachus and Penelope's "guests" are behaving very badly, alas, there are more than 100 of them in their house and Telemachus does not have the physical or political strength to drive them from the house. Thus, Telemachus states in Odyssey 2:
There is no one like Odysseus here to prevent the ruin of our house. We are not strong enough: we could only prove how weak we are, inexperienced in a fight. Yet I would honestly defend myself if I had the power: since things are done that cannot be endured, and the destruction of my house reveals injustice. (A.S. Kline translation)
Given Telemachus' comments here, it appears that the reason he and his mother need Odysseus is because he does possess both the political and physical strength to protect them from these unruly guests.