My Mom suddenly passed last November and while I love her and did love her, she wasn't "stable" enough to have a gun unfortunately lol.
However if I WERE to give her ( say things were simply so bad she had HAD to have one) the shotgun would have had to been .410 , 12 gauge is simply OUT of the question, and 20 gauge probably would have still been too much.
But with that fancy load of Winchester PDX-1 being made in .410 now, I would probably just get her a .410 with plenty of PDX-1 and maybe Remington HD Ultimate Home Defense shells.
And I'd probably get her the Saiga 410 for ultimate shooting ease, probably the softest shooting cause its a 410 Semi-auto gas operated, nothing to do.. I'd make sure she knows how to use it but basically would have her load it up, put the safety on, and keep it somewhere hidden but accessible. Also I would consider a Mossberg, I'm seeing a "500 HS410" Home Security 6 Shot 410 shotgun that looks like a mom type weapon.
Now everything I just said is of course purely hypothetical. I have no experience that I could actually claim with her, but it's what I WOULD have done if I was going to get her one.