I have this sentence 隣の席が私なのを、感謝しなさい
What does 私なの mean in this case? Why is there a を with a comma right after?
It's nominalizing. The sentence means "be thankful that the person in the seat next to you is me." But more literally, something like "be thankful for the thing/fact/way that, the seat next to you is me."
You could express the same thing as 私であること, but this sounds much more formal.