Telemachus and Penelope both occupy relatively lowly positions in ancient Greek society. Although Telemachus develops considerable maturity throughout the poem, as a relatively young man he still isn't strong enough to be able to take on his mother's suitors and expel them from his father's palace. For that, he will need the assistance of Odysseus. Hence the importance of setting off on his voyage to find his long-lost father.

As a woman in Greek society, Penelope is expected to be subservient to men. She doesn't like or admire any of the suitors, but she knows that at some point she'll have to give in and marry one of them. In the meantime, all she can do is stall them, using trickery to delay the dreaded hour when she will have to choose one of the suitors to be her new husband. But she too cannot remove the suitors from the palace; only Odysseus, with his physical strength and authority as king of Ithaca, can do that.

Penelope has little status and no power as a woman, though she does have great wealth. Telemachus, though he is now over 20, has never proven his manhood. In one respect, neither "needs" Odysseus. Penelope could agree to marry the least offensive of the suitors, and would thus provide Telemachus with a step-father. But that is not Odysseus' fate, and such a conclusion would have been repugnant to the Greeks.

Telemachus and his mother need Odysseus so that they can rid their house of the suitors that eat their food and drink their wine and waste their goods. Without Odysseus Telemachus' mother Penelope cannot get rid of them without choosing a suitor. Although she did not want to remarry any of them, with her husband gone for ten years, she has no choice but to consider them, and no way to get them out of her home.

In the Odyssey, why do Telemachus and his mother need Odysseus?

They need his strategic planning, his thirst for revenge, his patience, his ability to string his bow, and help from the god Athena to defeat the suitors.

Penelope has tried to trick the suitors with her cunning.  She has kept them at bay by weaving the shroud.  Her cunning is impressive, but the suitors are wise to it.  She has bought time, but her plan cannot make them go away.

Telemachus is too impetuous and brash to defeat the suitors alone.  They will kill him easily.  He cannot string his father's bow.  He does not possess the magic of Odysseus' cunning yet.

Only with Odysseus, can Telemachus defeat the suitors.  Penelope doesn't even know Odysseus is on the island.  She is not part of his plan.  Revenge is not woman's work.

Remember, Odysseus is not coming to the island alone.  He is bringing Athena with him.  She disguises him as a beggar.  If he shows his face, he will be killed instantly.  So, his plan of revenge is ordained by the goddess.  It can only be accomplished with help from the gods: this is the lesson that Odysseus has spent 10 years learning.

Odysseus rallies his son and his troops, the herdsmen.  He exercises cunning and patience in his plan.  He traps the suitors all in his hall, lures them there with a false promise: he who can string the bow will become king.  His plan of revenge is as cunning as his plan to end the Trojan War with the wooden horse.  After he strings the bow, his identity is finally revealed and he and Telemachus kill them in a river of blood.

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.

Why do Telemachos and his mother need Odysseus?

Telemachus is only a young man and so is in desperate need of guidance from a male authority figure. As the son of a king Telemachus will one day ascend to the throne, but with his father absent from his life he's unable to learn the vital skills that will stand him in good stead later on. The goddess Athena, Odysseus's protector, is a big help to Telemachus in giving him the courage to address the people of Ithaca before embarking on an epic voyage to track down his missing father. But it's not enough. Telemachus needs Odysseus at home if he's to learn the valuable arts of kingship.

As for Penelope, she needs Odysseus to get rid of all those bothersome suitors that have been paying court to her all these years. Telemachus isn't strong enough to do the job himself; only Odysseus can do that. For only Odysseus has the skill, strength, and authority to drive the suitors from the royal palace. So the sooner he arrives back on Ithaca after his epic wanderings, the better it will be for Penelope.

Why do Telemachos and his mother need Odysseus?

Telemachus is young, but he and Penelope have been overrun by suitors looking to marry Penelope and take all that belongs to Odysseus. Telemachus will have no rights if his mother marries another man, and Penelope is expected to choose another husband since Odysseus has been gone so long. Also, the suitors have stayed much too long and are taking advantage of Telemachus' young age. Both Telemachus and Penelope know that the only way solve these problems is for Odysseus to return.

Why do Telemachos and his mother need Odysseus?

Telemachus is a teenager and unprepared for the responsibility of a kingdom. As for Penelope, it a woman had no way to support or defend herself. No one is better suited for the job, in her opinion, than Odysessus.

Why do Telemachos and his mother, Penelope, need Odysseus to get rid of the suitors?

In the Greek society of Homer’s stories, hospitality was a primary quality of any household, especially a royal one.  It must be remembered that Odysseus before the Trojan War was the “king” or main leader of Ithaca, a city-state of immense wealth and power.  Also, widowhood (which everyone assumed was the condition of Penelope until some proof of Odysseus’ fate was found) was a socially uncomfortable condition, Greek women having very little decision-making power. Telemachos was not of age to take over his father’s position as leader of Ithaca.  The gradual overfow and advantage-taking of the household began as “friends” came to pay their respects, offer condolence, etc., but gradually Penelope’s natural hospitality, together with her wealth and beauty, turned the well-wishers into suitors, all too powerful for Telemachos to combat either physically or socially.  The only solution as Penelope delays her decision is for Odysseus to show up alive and reclaim his place in the household and the city-state.  The final passages in the epic, describing his homecoming, disguise, and final confrontation with the “suitors,” is arguably the most dynamic and exciting action/love/family story ever written.