What Documents Do I Need to Apply for an Extension Based on Having a Thai Dependent Child in Hua Hin?

Unless I've missed it being mentioned, the child's birth certificate (which should have your name on it).

nope, it takes the exact same documentation every year, photos of the child and you in front of the address sign at your place, inside the house, usually photos of them and you standing in front of the school they attend, map to your house, all the documentation showing you meet the financial requirements, have parental rights as a foreign father, copies of the passport, AND taking your child <- the reason you're getting the extension in the first place with you when you apply.

You'll get a 30 day under consideration stamp, they might come to your house and do an in home visit during that time, then you go back when that date is up and get the year extension inked into your passport

I was asked for dependency docs

That would be for a 60 day extension, not a year