I am doing some home improvement in the LR (plumbing and electrical). I need to open the wall up.

Is it better to cut the drywall section in the cavity, down the center of a stud, or adjacent to a stud?

Better here is measured about ease of attaching sections in the replace phase.

I am thinking of re-using the same sections to match texture. We'll tape, mud, texture (if needed) and paint, but by no means do we spend enough time in there to really worry about small detail.

Thanks.

I'll answer that question with another?

If you cut the drywall adjacent to the stud, how are you going to attach it back?
Screws that attach to air?

You will have to provide a backing for horizontal cuts as well, whether it be a full 2 x 4 or a say a 1 x 3.

My preferred way is to saw the drywall adjacent to the stud, using the stud as a guide. The only way I know of cutting down the center of the stud is with a knife and that takes to long. To reinstall I tack or screw a nailer (1x2 or 1x3) to the side of the stud and attach the patch to that. I see no reason to reinforce the horizontal joint. That joint is no different that any other drywall joint spanning the studs. Nothing wrong with reusing the cut out material to patch if it is still intact by the time you are through with your other repairs and you can find it. I don't understand what you mean by matching texture. If you have textured walls you are going to loose the most of the texture on the patch piece when you tape and mud unless you have very large cut outs. The texturing will also make taping difficult. If you have textured walls you are probably better off replacing the cut out with new material, scraping away the texturing around the patch then taping and mudding. Then re-texturing the entire area.