The question really is, how many ports does the Mac Mini really need? HDMI certainly to keep its usefulness with HDTVs (though the loss of Front Row sucks), at least one Thunderbolt port, three or four USB ports, power connector, it only really needs a single audio connection, but what else? Ethernet is nice to have, but with built in wireless not everyone uses it, and it's easy to add via Thunderbolt; I wouldn't like it, but it's how Apple may see it.
With minimal ports the Mac Mini wouldn't really need to go vertical, and likewise reducing to a bare minimum the ports hardly need to take up any space at all. The rear ports on a Mac Mini have always been awkward, to the point that you don't really
need more than two USB ports, such as one for a backup drive and one for a wired keyboard with integrated USB hub, or a separate USB hub if you use a wireless keyboard. Over time more displays will hopefully replace Display Port with Thunderbolt in order to add extra ports (which are more convenient anyway).
For example, here's a really quick and dirty re-arrangement of ports; image sucks a bit, but it shows how they can be reasonably compressed, and should fit pretty well even on a circular profile. For cooling the system could use vertical slits on just the right side, angled backwards so they're hidden from most angles.
Without ethernet and only a single Thunderbolt port (might be pretty likely for a smaller system) you could move the Thunderbolt over to the HDMI port. So yeah, there is absolutely room to do this, and the above image is actually keeping flexibility, I wouldn't be surprised to just lose the Firewire port with no extra Thunderbolt, and maybe even lose a USB port or two as I'm not sure Mac Minis desperately need them.