Native speakers would say
... him being only eight
... he being only eight
probably 80%/20% in favor of him being only eight, possibly 90%/10%.
You will probably find some older textbooks insisting that he being only eight is the proper form and him being only eight is non-standard or colloquial. You will find he in more formal texts, and him in conversational texts and in literary works that realistically reflect vernacular speech.
P.S. Since your example sentence includes y'know it would be no surprise to learn that the textbook expects him there.
P.P.S. What does not work there (with OP's sentence "as is") is his being only eight.
It's still light out, but him being only eight, we put him to bed.
He was booked on a manslaughter charge, but he being only eight, perhaps the DA will offer him a plea deal. Some hardliners want to know how his being only eight is relevant.
P.P.P.S. Now that we know it is a transcript of a conversation: his being only eight is IMO a "hypercorrection". Absolute phrases, which are sentence modifying clauses, are formed with the nominative + being or the accusative + being but not with the genitive + being. However, for many decades prescriptive grammar textbooks taught that accusative + being was wrong in certain patterns, and so now some speakers in all circumstances say his being instead of him being as a result, just as they always say "he and I" or "she and I" in situations calling for "him and me" and "her and me", because they've mislearned their grammar lessons; these are hypercorrections as well; they're not "natural".