I would say “he was born into a rich family.”

I also agree that “Born of”is better in this sentence.

Of these, I would choose 2. of. "He is born of rich parents." However, I think a more typical sentence would be "He is born to rich parents." English plays fast and lose with prepositions a lot of the time, so it's hard to really explain why one over another. In this sentence, it is definitely not "in," as being born is literally coming "out." It is not "by" because of your verb phrase. If you changed it to "He is birthed by rich parents," that would work grammatically, though it is still not the preferred phrasing, as that emphasizes the mother's labor, and it seems like the point of the sentence is that both parents are rich, not that the baby was born. Arguments could be made for "from," as children are made from their parents, but the best word in my opinion here is "to."

Answer has to be "of".

in