Who will rule the island when Ulysses retires?
It becomes clear as Ulysses sets off on his journey that he feels he leaves Ithaca in capable hands. Ulysses believes that it is in his best interests to leave his son, Telemachus, as the leader of Ithaca when he sets off on his journey: "This is my son, mine own Telemachus,/To whom I leave the sceptre and the isle,—" Ulysses feels that Telemachus will be a good ruler for Ithaca and thus leaves him to rule when he retires.
Tennyson has Ulysses describe him as "well loved" and the father says that his son is capable to "fulfill this labor." Ulysses believes that Telemachus possesses a domestic capacity that enables him to be successful in taking over for him as the day to day ruler of Ithaca:
Most blameless is he, centred in the sphereOf common duties, decent not to failIn offices of tenderness