What is the process for extending a tourist visa in Phuket, and which immigration office is better?

For the regular 30 day extension either office will work.

Usually avoid Mondays and Fridays as those days seem busier, but post on a phuket group and ask the best days to go.

There's no need to wait until your current stamp is running out as you can get an extension when you have 30 days or less left on your current stamp and they'll add the 30 days to your expiration date.

You need copies of the data page of your passport, the visa, the entry stamp and your departure card along with a passport sized photo and a filled out TM7 application for extension along with 1900baht.

There are copy places near BOTH immigration offices down there in Phuket.