The bow is a symbol of kingship and strength. Whoever strings it is worthy of King Odysseus and can rightfully take the throne. Physical strength was an important quality to the ancient Greeks, not only in warfare, but because psychologically it was tied to political strength and the iron will needed to govern a people who were given to argument and debate.
Not surprisingly, none of the suitors even comes close to succeeding, showing their weakness and folly in vying for Penelope’s hand. Telemachos almost strings the bow and might have done it had Odysseus not interrupted him; this shows that Telemachos has matured into manhood and is almost ready to take the place of his father. Of course, that he stops upon request reminds us that he maintains deference to Odysseus’s will.