Where do you cut drywall if you need to remove it but still want to easily put it back up?
I'll deviate from the other suggestions. I prefer to cut in open space and float new backing.

It's more difficult to cut down a stud. The knife or saw can't penetrate cleanly through, and you tend to hit fasteners.
If you're too far off the stud center you'll have a hard time putting screws back into the skinny overlap.
New, floated backing can be wider than a stud. This makes re-fastening both the patch and the edge of the existing drywall easier. 2-3&quot; wide strips of 1/2&quot; plywood are great. So is scrap 1x4 or whatever you have on hand thick enough to hold a screw.
The new backing doesn't need to be mounted to framing. Fastening blocks to framing and keeping them flush is often a challenge, especially with small openings. Floated backing is automatically flush with the framing.
New backing can be floated beyond the patch and screwed into the existing drywall. This can provide needed support in some situations.
It provides flexibility of location. I don't always want to cut on a stud because of where the taping patch needs to be.
The cutout can be whatever size or shape I like. If I'm reusing the piece, it's easy to float random backing behind the edges of an oddly-shaped cutout. If I'm installing a new patch, I cut that first, trace it on the wall, and cut the trace. Perfect fit every time.
New backing can be installed over insulation, often making the job easier.
Patch size doesn't matter. If it's large enough you'll have some studs to use as well, but this method is great for the edges of the repair.

Of course, one solution doesn't apply to every case. Use what make sense.