Well... yeah. The sentence is built as "This is Y", right? And the noun phrase Y is "my neighbour's son". If we consider this noun phrase on its own, it's "syn mojego sąsiada". Saxon Genitive in English = Genitive in Polish. Now we're putting this Y into the sentence and we see that Y must take Instrumental. And luckily the Genitive part doesn't change anymore here, you just need to put "syn" in Instrumental. "synem mojego sąsiada".
Respectfully, I'd argue that it is:
Respectfully, I apologize, seems that you are right. However, I'd still prefer to keep the translations as literal as possible, especially that you're really the first person to suggest that. As 'just acceptable answers' (not the main, starred ones) sometimes get suggested without any particular reason, I don't want anyone to get such a suggestion and get confused. Especially people that are not native speakers of English could get confused about the exact meaning of "syn".
Ok, so I'm confused, yet again. Lol Sorry. So this sentence translates literally to: "He is the son of my neighbor". Is my observation correct that in Polish, possession follows this order: thing that is possessed, followed by word like my or yours that shows who it belongs to, and lastly the person, thing etc that it belongs to?
You're generally correct, but I'd say that there are rather two notions here, not three. First is 'the thing that is possessed', but the pronoun "my" and the noun "neighbor" together create one notion, "my neighbor", and that is 'the person that it belongs to'. This sentence has "my", but it could also be "He is Adam's son" - "On jest synem Adama".
The 'owner' always takes the Genitive case, the 'owned thing' takes the case that is needed in the given sentence.