Shotgun? Come on. Unwieldy, she can be easily disarmed, recoil will knock her on her butt. Also not something you carry around the house, you know what I mean? Still, effective in the right hands, of course, but not hers.
All valid points against the shotgun, except for the last sentence. However, just because they are valid points does not mean that they apply in all situations, to all people. My mother, at 4'11" (and a half!) and wearing a size 3 1/2 ring could manage a 12 ga SxS, and even Dad's S&W M28 (6") .357 Magnum. See actually preferred the shotgun, and she did NOT fire it from the shoulder! She managed the recoil firing from the hip, (as she said "like Victoria Barkley") letting the gun swing back and just holding on. This is plenty good for across the room, and especially if the intruder is in a doorway.
Mom' pistol was a Ruger Super Bearcat, which she was quite good with, and with a shotgun available for her piece of mind, she was content she could handle any situation. OF course, I also saw this woman face down and corner a 900lb Angus bull with a pitchfork, and hold him at bay for an hour until Dad got home! It is all about attitude and determination, the will to use what you have, and never give up.
IF you don't have a shotgun, get one. .410 or 20ga will do fine, 12ga only if it seems possible and practical for her. Keep it somewhere in the house where she can get to it. A short barrel pump or a SxS.
The .22 pistol is not to be overlooked. A .22 she can use is better than a .32 (or anything else) that she can't. A single action revolver in .38 cal is worth considering, or teaching her how to use both methods of shooting a DA revolver. Shooting DA is for the "no time to do anything else" situation, and even though she may have trouble in practice, I'm sure that in the real and scary, she will have the strength to pull that trigger as often as needed, if she has the will.
Home invaders come in 3 general catagories, those who are only looking to steal, those who will steal and are willing hurt/kill to do it, and those who are looking to hurt/kill for their own pleasure. We plan for the worst, and hope for the best. The worst has to be stopped. If they die as a result of being stopped, so be it. Most intruders can be driven off by a shot, even if they are not hit, but you cannot count on it for certain.
Discuss it, and decide on a retreat room, one you can barricade (locking door). Phone (and gun) in that room, at least. And that door doesn't get opened until the police ask her to (and they are proven to be the police!). Seeing the squad car out the window, or getting the officer's name and having it verified by 911. Bad guys have been known to lie. Some even have fake badges.
Get her comfortable with some kind of gun. Get her comfortable with what she might have to do, and how to do it for her safety. Everything in your house can be replaced or repaired, except her. Don't take chances.
Good Luck.