Many good ideas have been posted so thought I would throw mine in also.
The way we do it in the business of repairing may not be the way everyone else does it, but we have found it to be easy and strong.
We either cut a piece as close as possible to the hole or we will cut it the hole into a square and cutting a piece that size. We next attach a piece of 1 by (X) approx 3" larger than the replacement piece (sheetrock screws). Then insert the piece into the hole (put screw in the center to hold it in place) and attach the piece of 1 X on the ends to secure the piece into the hole and then just proceed as one would apply sheetrock mud until they reach final finish.
