it'd be nice if I could find a case that would bolt to the underside of her desk for space-saving purposes

Hmmm. I just realized there's nothing stopping me from taking all the components out of the PC and bolting them, individually, to the underside of the desk. Spread out over the entire surface area so that there's no ventilation issues. Done properly, you could do away with most or all of the cooling fans and have a silent, caseless PC.

Of course, you'd still need that spare Shuttle machine to take to LAN parties.