Why not get both a dog and a shotgun for her?

I believe that the dog would be sufficient to handle most situations. If, however, someone were to get past the dog, she'd have something more substantial to protect herself with.

People can sometimes be brave and slip past the largest of dogs sleeping in your yard, but I'm pretty sure that they'd take a small woman with any shotgun REALLY seriously.

The key to shooting accuracy is practice AND feeling at ease with using the weapon you buy for her. If she lacks the confidence or is scared by the gun, you MUST overcome that fear before proceeding. Shooting blindly/with eyes closed/or a severe shake just doesn't work. Start her with a small .22LR. Let her fire off a brick at the range. Get her comfortable handling the gun. The confidence will come. When done, move on to the next weapon. Perhaps something like a 9mm carbine or something that fires a .223 round? A .410 and/or birdshot isn't sufficient, especialy for last ditch self defense.