In my experience it’s a bit old-fashioned (my grandmother was fond of it but I wouldn’t use it) but I think it’s pretty well known. You use it when there is something that you think you might be able to answer, but can’t answer right now. It often has the idea of making an effort to solve a problem. You want to join my class but you didn’t sign up by the deadline? Let me scratch my brain. (I’ll try to think of a way) A: Who was the lawyer we used when we made our will? B: That was decades ago! I’ll scratch my brain and get back to you.

I'm a US speaker. I've never heard "to scratch your _brain_." "To scratch your _head_" is an idiom for "to think hard and long." When you are thinking about something, it is not unusual to fiddle around without your hands, and one of the common things people do is to scratch their heads. That is, _literally_ scratch their heads. To "scratch your brain" sounds like someone making a joke by slightly changing the standard phrase." I agree with others: I "rack my brain" and I "scratch my head" but I don't "rack my head" or "scratch my brain."

There is no exact phrase "to scratch your brain" but there is the adjective "mindbending", and the noun "brainteaser". "The movie The Matrix is a mindbending brainteaser."

Hi Gareth, I am a British native speaker and I have never heard this expression. Maybe it is American. I don't know. We would say "rack my brain" when we think hard. "I've been racking my brain but I can't think of where we can go this weekend". I hope this helps. Victoria

Wait whilst I scratch my brain for the answer!