Learn Turkish: Onun alacağı hatırayı bilmiyorum ama onlarınki pahalı olacak. - I do not know the souvenir he will buy, but theirs will be expensive.

The word onların is the genitive pronoun meaning 'their'. By adding -ki, you create the possessive pronoun 'theirs' (literally 'the one belonging to them'). Here, it replaces the phrase onların hatırası (their souvenir) so we do not have to repeat the noun.

Since the lesson is about -ki, why isn't it onunki at the beginning of the sentence?

Because onun is simply acting as a possessive word for the phrase that follows it (onun alacağı hatırayı). You only attach -ki when you want to drop the noun completely and say 'his' or 'theirs' as a standalone pronoun, just like we did with onlarınki at the end.

Why is there a possessive ending on alacağı?

This is a relative clause. To say 'the souvenir that he will buy', Turkish takes the verb almak, adds the future participle -acak, and then requires a possessive ending to show who is doing the buying. Since it is 'his' (onun) action, we add -ı (which softens the 'k' to 'ğ'), making alacağı.

Why does hatıra have the -yı ending?

It takes the accusative case ending -(y)ı because it is the specific direct object of the verb bilmiyorum (I do not know). You are stating that you do not know this specific souvenir.