What documents are needed to extend a non-immigrant O visa for dependents of a UN employee in Thailand, and when can we apply?

You can't be the first employee of the UN that needs to have their family get matching extensions based on your working here.

You need to have them call the immigration office and get the list of required documents for their extensions OR you need to go out to the immigration office and get the list yourself. There is no one size fits all list of requirements. Obviously you're going to need the marriage certificate, birth certificates (now whether they needs to be translated into thai or not I dunno), letter from your company stating it's okay to have your family get extensions, your visa, extension, contract, work permit etc

BUT I'm just guess, you need to contact or GO to the immigration office to get the real list.

In Bangkok you can apply for an extension when you have 45 days or less left on your current entry stamp