This is my son, mine own Telemachus, 

To whom I leave the sceptre and the isle,— 

Well-loved of me, discerning to fulfil 

This labour, by slow prudence to make mild 

A rugged people, and thro' soft degrees 

Subdue them to the useful and the good. 

Most blameless is he, centred in the sphere 

Of common duties, decent not to fail 

In offices of tenderness, and pay 

Meet adoration to my household gods, 

When I am gone. He works his work, I mine.

It is important to realise that Tennyson is using the Odyssey by Homer to write this poem. Ulysses is the Latin name for Odysseus, one of the Greek heroes who fought in the Trojan War and was actually responsible through his cunning and guile for conquering Troy and ending the conflict through his idea of the Trojan Horse. Homer's Odyssey tells the tale of the return of Odysseus back to his beloved wife and son, to take up his rightful rule of Ithaca once more. However, in this poem, Ulysses is now imagined as an old king, desiring one last adventure before his ultimate death. It is clear if you read the poem carefully that Ulysses plans to leave the kingdom in the safe and capable hands of his son, Telemachus. Note what Ulysses says:

This is my son, mine own Telemachus,

To whom I leave the sceptre and the isle,--

Well-loved of me, discerning to fulfill

This labour, by slow prudence to make mild

A rugged people, and through soft degrees

Subdue them to the useful and the good.

Ulysses thus presents his son to the population of Ithaca as a worthy replacement, as a man who will work hard at his job of ruling them and will patiently maintain stability and encounter every problem that could arise.