Hi all... Just as a follow-up, I wound up getting the iFixit dual-drive upgrade kit mentioned above and putting a Samsung 250GB SSD in there (I also swapped out the original HDD for a 2TB model while I had the guts of the mini open).
I definitely agree with @chefklc that the install procedure is not for the faint-hearted. I actually had a lot of fun doing it because it's been a while since I had to do that much physical work on a computer, and it gave me a lot of appreciation for the really outstanding work Apple put into the hardware design of the 2011 mini. But it's a pretty complex procedure
during which every single component of the machine is removed from the case, and unless you get a great deal of enjoyment out of doing that kind of thing, I'd echo the sentiment that going with an external ThunderBolt drive is probably far easier.
Beyond just the physical install, getting OS/X onto a bare drive in a mini with no boot media is a profoundly irritating process during which the mini will download an entire install disk from Apple every time you boot. I finally realized that I could use the recovery partition from the original drive to do it and that sped things up somewhat (I had ordered an external 2.5" USB enclosure which was useful for this). I had already backed the mini up with Time Machine before the surgery, and everything came back perfectly, once I was finally able to get the recovery OS/X thing going.
I also ran into another problem while trying to install Boot Camp on the mini - I hit the dreaded 0x80300024 error
and the Windows 7 installer wouldn't install anything in the partition I'd created. Fortunately, it wasn't too difficult to disassemble the mini enough to be able to pop off one of the two drive controllers and then continue the install (although, inevitably, I took the wrong one off and had to shut down, open the case again, replace the controller I'd disconnected and attach the other one, then put everything back together and start from scratch).
[Edit - to clarify the above, if anyone is curious, some Windows 7 install media will run into problems installing the OS onto Macs (and maybe other types of PCs) if more than one internal hard disk is present. Thanks Microsoft! The workaround is to disconnect one of the two drives, install the OS, and then reconnect the drive you disconnected once Win7 boots properly.]
So far the thing works like a charm - much faster and much quieter than my old HTPC. Thanks again for the advice!