I think the ultimate souvenir would be not a moon rock or a martian, but space "nothing", but I wonder if this is physically possible... Can an astronaut go out to space with an open container, close and seal it and bring it back to Earth? Is there a container that could withstand the pressure of Earth's atmosphere upon reentry? If this is possible, what does my container of space "nothing" truly consist of?
Space is not quite as empty as you might think, and your average jam jar is easily strong enough to hold atmospheric pressure against an internal vacuum. More to the point, there wouldn't be any need to go into space to create a jar full of nothing. Current technology can easily create a vacuum that would compare with that of "outer space".
As for what your container will contain, it depends on your point of view. In everyday terms, there is nothing, except perhaps a few stray particles bouncing around. If you agree with the relevant underlying theory, your container is full of quantum foam. But then, the quantum foam is everywhere, and the container just happens to be empty of anything else.