I think it would be a better idea to have a keyboard-box (PC inside a keyboard). You would have more space to work and it doesn't have to move due to its very nature.
A keyboard the size of the Apple wireless keyboard would be excellent ( https://www.apple.com/keyboard/ ).
Of course now we have Penny's "computer book" from Inspector Gadget so it's not quite as mindblowing as it was for 8-year-old me.
This was my first reaction, too, but I'm not sure it makes sense, at least not categorically. My current Logitech mouse is over two years old with no issues at all. And I don't know that I'd expect a tiny ARM computer with no expansion to be worth using on general purpose software more than a couple years. Still probably wouldn't trust a newcomer to produce a mouse able to go the distance and worth keeping for years, though.
I hate wires coming out of my mice. I can tolerate the thin braided cables coming out of most high-end mice, but that's about it. With Mouse Box, I'd have display, USB, and a bunch of other stuff sticking out of the mouse making it much harder to move around and use.
The mouse itself looks unexceptional. It's not especially ergonomic (my Logitech MX Revolution looks much more comfortable to hold), and the button set is pretty limited. It's fine for basic use, but it would kind of defeat the point if I felt compelled to plug a mouse into my Mouse Box.
Mice wear out. Mine collect dust and scuffs on the bottom and sweat and dirt on the top. I guess the mouse itself could conceivably outlive the usefulness of the computing part, but it seems like a bad idea to put the computer in a consumable device.
The use cases seem a bit contrived. I might accept the projector one because clicking to advance through slides is something that people do, but surreptitiously using it at work by connecting it as a second input to the monitor sounds pretty far-fetched.