Questions & Answers about Das Geschenk ist für ihn.
Why do we use ihn instead of er here?
The preposition für (for) always takes the accusative case. Since the gift is for a male person (him), we must change the basic he pronoun (er) into its accusative form, which is ihn.
I thought the verb sein (to be) takes the nominative case. Why is there an accusative pronoun here?
You are right that sein (here: ist) usually links two nominative things, like in Er ist ein Mann. However, in this sentence, the pronoun comes directly after the preposition für. Prepositions act like the bosses of their own phrases: für strictly demands the accusative case, which overrides the usual rule for the verb.
What would the sentence be if the gift was for a woman?
You would just swap ihn for the female accusative pronoun. The nominative sie (she) stays exactly the same in the accusative case, so the sentence becomes: Das Geschenk ist für sie (The gift is for her).
Is Geschenk a neuter noun?
Yes, it is das Geschenk. A helpful tip for remembering noun genders: many German nouns that start with the prefix Ge- are neuter, though you will occasionally find exceptions.