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.
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.
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.