That’s big enough to not just use speckle to cover it up. I’ve used the 3M small hole kit before and it has everything you’d need to cover the hole and is easy to use: 3M High Strength Small Hole Repair Kit with 8 Ounce Spackling Compound, Self-Adhesive Patch, Putty Knife, and Sanding Pad https://www.amazon.com/dp/B00IOQKUQG/
You’d need to get the right paint to match your wall, but if you only need to cover it, this kit is easy to use.
> What is the best method to patch drywall that has (roughly speaking) a fist sized hole in it?
Drywall patch. You can find it at your local Home Depot or Lowe's. It's easy enough that any schmoe can do it. Conversely, you can order something like this which has everything you need in one kit. Make sure to pick up some touch-up paint if the wall isn't white.
You'll end up with a small hole in the sheet rock which can be patched -- likely a good idea if you are not planning to hang another picture over the spot from which you remove the picture hanger.
There are other ways to patch the hole as well, including this one ... but ... the larger you make the patch, the larger the place will be on the wall that "begs" to be repainted:
https://www.amazon.com/3M-Patch-Primer-Self-Adhesive-Sanding/dp/B00IOQKUQG
its really easy to fill them. we did it with this:
https://www.amazon.com/3M-Patch-Primer-Self-Adhesive-Sanding/dp/B00IOQKUQG
just sand, fill, and sand again when its dry.
Thanks for the suggestion. What should I use to patch the drywall? Will this work?
I used whatever comes in this kit.
What's the appropriate width of knife? I can get another putty knife.