Yeah, I know the problem. I use a very light mesh, much more fragile than the heavy no-see-um, and holes happen - snags or animals.
> cut the square so that it is larger than the tear by roughly 2", and then either hand-sew the patch
That's what I do, although I probably only allow 1" of overlap. I pitch the tent (actually, I often do the repair in the field anyhow), pin the patch to the fabric, then use a long needle. I run the needle in and out of the material several times before pulling the thread through - makes sewing fairly fast. I usually do a loop or two at the corners so the thread does not get pulled tighter than the mesh.
I tried glue and double-sided tape: they didn't work that well for me.