This is the Domesday Trackerball. It is the same model as the Marconi ones I have, except that it is "Acorn" badged and has labels for the "Action" and "Change" buttons.
I was lucky enough to have a bit of a play about with this Domesday System. It was the first time I had used one since mine was dismantled in January 2007 so it took a while to remember how to navigate around the Domesday Discs!