Learn Finnish: Minä ostan kahvia heidän serkulleen. - I am buying coffee for their cousin.

In standard Finnish, third-person possessive suffixes (like the -en ending here) require the genitive pronoun hänen (his/her) or heidän (their) before the noun. You can only drop heidän if the subject of the sentence is the same 'they' who owns the object, but here the subject is minä (I).

Where does the -en ending in serkulleen come from?

It is a third-person possessive suffix. The base word is serkku (cousin) in the allative case serkulle (to/for the cousin). To say 'their', we add a possessive suffix. Instead of using -nsa/-nsä to make serkullensa, it is very common to use the -Vn suffix, which means doubling the final vowel and adding -n. So, serkulle + e + n becomes serkulleen.

Why is kahvia in the partitive case?

Kahvi is an uncountable mass noun. Because you are buying an unspecified amount of it ('some coffee' rather than 'the coffee'), the object of the sentence must take the partitive case.

Why does 'for their cousin' use the -lle (allative) ending?