Past the statute of limitations; he's dead; and I'm sure we can agree that Neil f*ing Armstrong can keep a memento/be forgiven for overlooking a memento or two. And how do you know no one said "shit Neil, keep it if it means so much to you". I know I would have.
I don't think that's what happened.
The original story goes that Buzz Aldrin was supposed to be the first one to walk on the moon, but during the trip, an order from mission control came in that said that Neil Armstrong was supposed to be the first. Buzz Aldrin had been politically outmaneuvered. And Buzz was so upset about that, he retaliated by supposedly 'forgetting' his own camera on the moon. Obviously, most of the shots on that camera were shots of Neil Armstrong, so most of the photographs we have now (except for the video feeds from the lander) are actually pictures of Buzz Aldrin, not Neil Armstrong.
So upon hearing that Buzz Aldrin had accidentally left his camera on the moon, it would certainly make sense that Neil Armstrong would grab something as well. In any case, it would be really cool if Buzz Aldrin had also lied about not bringing the camera back with him, but in his case, it's actually less likely. If I had been Buzz, I would have also left the camera on the moon in the rage of the moment.