I think that part of this song means that when they went back to school or home or whatever their normal life was that even though things got back to the way they used to be, with all of their teachers giving work & tests and the norm or when their parents bugged them to get a job (they can take, take, take, take the kids from the summer) they still remembered all of the memories and times from the summer.
And then he's asking summer for one of those memories. He's asking for summer to give him something to remember, so that when everything goes back to the way it was, he has something to think about.