Learn Russian: Родители назвали сына Иваном.

In Russian, verbs of naming or calling, like назвать, require the name being given to be in the instrumental case. Since the base name is Иван, it takes the standard masculine instrumental ending -ом to become Иваном.

Why is the word for son written as сына instead of just сын?

The word сын is the direct object of the verb, receiving the action of being named, so it must be in the accusative case. Because a son is a living person, it follows the animate accusative rule where the form matches the genitive case, adding an -а.

Since the current lesson is about the prefix пере-, how would I say the parents renamed their son?

You would use the prefix пере-, which indicates doing an action over again or differently. Adding it to this verb gives us переназвали (renamed). The rest of the sentence remains exactly the same: Родители переназвали сына Иваном.

Is назвали perfective or imperfective here?

It is perfective, emphasizing the completed, one-time result: the parents successfully gave him the name. The imperfective pair is называть. If you used the past imperfective называли, it would change the meaning to a repeated or ongoing action, meaning they used to call him Ivan.