First boon

Having done worship to naciketa, yamaraja said:

tisro rātrīryadavātsīrgṛhe

me‘naśnan brahmannatithirnamasyaḥ |

namaste’stu brahman svasti me’stu

tasmāt prati trīn varān vṛṇīṣva ||1.1.9||

Oh brāhmana, who is a worshipful guest stayed in my place for three nights without eating. Let there be salutations unto you. Let there be well-being unto me. Therefore, may you choose three boons (for each night).

Yamaraja being a great person, with humility did a salutation to naciketa - namaste’stu as a brāhmana guest who is worthy to be worshiped - brahmannatithirnamasyaḥ. Even though he didn’t deliberately commit such mistakes as neglecting his guest, he still asks for forgiveness from naciketa for causing him to stay at his place without food for three nights. He also asked naciketa for blessing - svasti me’stu, by choosing three boons - trīn varān vṛṇīṣva for additional propitiation for naciketa to stay without eating for each night - tasmāt prati.

Since yamaraja desires of giving, naciketa asked:

śāntasaṅkalpaḥ sumanā yathā syād

vītamanyurgautamo mā’bhi mṛtyo |

tvatprasṛṣṭam mā’bhivadet pratītaḥ

etat trayāṇām prathamam varam vṛṇe ||1.1.10||

Oh lord of death, may gautama (the name of naciketa’s father) be free from anxiety, cheerful, and not angry toward me. When I am released by you, he should recognise me and talk to me. I choose this boon as the first among the three you have given.

Naciketa chose the following benefits which is the form of father’s well-being, as his first boon among the three. śāntasaṅkalpaḥ - one whose mind is subsided from worry/anxiety, since his father must be worried about naciketa’s going to yamaloka. vītamanyuḥ - his anger subsides, because naciketa annoyed him and put him in a difficult situation. Not just that, he should be sumanā - one whose mind is in good condition (cheerful and calm). Moreover, father should recognise - pratītaḥ and speak to him - mā’bhivadet when he is released by yamaraja - tvatprasṛṣṭam

After naciketa asked for the first boon, yamaraja told:

yathā purastād bhavitā pratītaḥ

auddālakirāruṇirmatprasṛṣṭaḥ |

sukham rātrīḥ śayitā vītamanyuḥ

tvām dadṛśivān mṛtyumukhāt pramuktam ||1.1.11||

Your father will recognise you as he was earlier. Being blessed by me, he will sleep happily at night. Having seen you released from the hold of death, he will be free from anger and talk to you.

Yamaraja ensures naciketa that when he comes back from yama-loka, his father will recognise him - bhavitā pratītaḥ and has the same disposition as before - purastād which is endowed with love/affection towards him. Being blessed by yamaraja - matprasṛṣṭaḥ, auddālakirāruṇiḥ (names of naciketa’s father) will have a calm mind to sleep happily at night - sukham rātrīḥ śayitā. And he would be free from anger - vītamanyuḥ, having seen naciketa being free from the hold of death - tvām dadṛśivān mṛtyumukhāt pramuktam.