Yeah, that should be entirely possible.
Here's the iFixit guide for replacing the original HD:
In combination with the steps to instal the second hard drive, again located here:
The final step of the second hard drive installation addresses the question of running the SSD as the primary. Specifically it says:
"To minimize the amount of work necessary when adding an SSD to be used as your primary drive, we recommend installing the SSD like we did in this guide and leaving the stock platter drive in its original bottom position (closest to the antenna plate). After reassembly, boot the Mini into Lion Recovery and use disk utility to erase your SSD. Then install Lion on your SSD. After rebooting, go into system preferences>sta
Then you should be made in the shade. Since the device can boot off the SSD as a primary then you don't need the other at all.
Once you have the system booting off the first SSD, then you can go back in and replace the original HD with your second SSD.
Though you will be required to install them in two waves. Put in the first and get it to successfully boot off of the SSD as the primary. Then go back in and install the second SSD by replacing the original HD.
There is enough room within the device to install a second hard drive, about a centimeter and a half of extra space.
Which is documented in iFixit's Teardown of the device, located here:
They also sell a convenient tool kit specifically for this purpose. Located here: Mac Mini Dual Drive Kit If you're interested or in need.
Oh, and here is an installation guide as well:
Hope that answers your question.
Add a second hard drive to your 2011 or 2012 Mac Mini. Kit includes: SATA Hard Drive Cable, Mounting Grommets, Mounting Screws, Logic Board Removal Tool, 26 Bit Driver Kit, and ESD-Safe Spudger. Compatible with 2.3, 2.5, or 2.7 GHz 2011 Mac Minis (Model A — 50+ available at 29.95 each.