What is the process for extending a 60-day visa to a Non-O visa in Thailand?

Okay, what you're doing is a three step process (at minimum)

You will apply to convert the 60 day tourist visa to a 90 day Non-O visa by meeting the financial requirements

The application will go under consideration for between 14 and 21 days (depending on the immigration office) then you go back and get the Non-O visa and a new 90 day stamp inked into your passport

When that runs down to 30 days or less you will apply for a year extension, again by meeting the financial requirements and providing the appropriate documentation.

GO to the Samui Immigration office and see if they can do the tourist visa to non-O conversion (some offices cannot) and if they can find out how many days you need left on your current stamp to do it AND get the hand out that lists the requirements.

Usually the last 30 days of your current stay. But could be the last 45 days. Samui immigration will have that piece of info.

The problem is that you'll have to convert the tourist visa to a non-o visa first. Some offices do it, some don't. I don't know if Samui will perform that conversation. If nobody is familiar with this, you might have to ask immigration yourself.