Learn Finnish: Pienikin halaus voi tehdä surullisen ystävän iloiseksi. - Even a small hug can make a sad friend happy.

This is the core of how to express causing a change of state in Finnish. When the verb tehdä is used to make someone or something into a new state, that resulting state is put in the translative case, which ends in -ksi. Here, the friend becomes happy, so iloinen becomes iloiseksi.

Why are surullisen and ystävän in the genitive case here?

They form the total object of the verb tehdä. Because the action is considered complete and affects a whole countable thing (one friend), the object takes the genitive-looking accusative ending -n. Notice that the adjective surullinen must agree in case with the noun it modifies, becoming surullisen.

I thought the suffix -kin meant also or too. Why is it translated as even in pienikin?

You are right that -kin usually means also, but it frequently means even when emphasizing something surprising, extreme, or minimal. When attached to an adjective describing a very small amount, like pienikin halaus, it translates much more naturally to even a small hug.

Why does surullisen match the case of ystävän but iloiseksi does not?