If the monitor has VESA mounting screws you might consider a VESA mount. There is a replacement bottom cover you can buy that's VESA compatible, though it's pretty expensive.
For one of my own Mac Minis I just drilled holes into the cover myself (after buying a $25 spare, just in case it went horribly wrong). I just took the cover off, made a template for the 100mm apart holes, and found some screws that fit the thread of my monitor and were relatively flat; it was then just a simple case of screwing the bottom cover onto the back of the monitor, then twisting the Mac Mini into place. The most difficult part really was finding a way to securely hold the cover in position while drilling, as it's a very awkward shaped being tapered and round
Of course with the DIY option there's no locking of the cover, so you have to be carefully you don't accidentally twist the Mac Mini free of it. However, pressing the power button pushes the Mac Mini in the cover-closed direction so it hasn't actually been an issue, plus tidying the cables out the back (now the bottom) with a nice tight cable tidy also restricts any accidental movement.
If you're not confident about drilling the cover then I wouldn't do it, but it worked great for me, and saves a ton of space, with no apparent cooling issues for light use. Also, like I said, the price of a replacement cover is only about $25 plus shipping, which is a lot less than third party covers, though of course the latter are purpose designed for VESA mounting, so it's up to you.
