I'm going to refuse to give my cousin english lessons
Why are you going to refuse your cousin english lessons?
Because they want me to teach them for free.
It's understandable to not want to work for free. Teaching a language is a lot of work.
Good, I hope they will be able to speak fluent English soon on their own.
You are right to not want to work for free even if it is family. There is a lot of time and effort that goes into teaching someone another language. Maybe you can discuss compensation before just saying no.