The problem with a pistol is that it is very difficult to hit your target, especially in the middle of the night when you are scared and adrenaline is pumping and someone is in your home.
The other problem with a bullet is that, if/when you miss, the bullet can go through several walls and still be lethal.
The many follow-up shots in a semi-auto pistol are an advantage, but there is the disadvantage that if you miss 7 or 8, that is 7 or 8 bullets tearing through your walls. Many smaller calibers also do not have the stopping power needed to down an intruder that is crazy or on drugs. Mel Tappan recommended nothing smaller than a .40 caliber. Read his Survival Guns for details.
The shotgun gives you a much larger target pattern. You can miss a little and still hit your target. When you hit, the grouping of that light shot still has all the impact needed to down any intruder. If some of the shot misses, it will lose much of its power hitting wallboard (buck shot will still retain quite a bit). If you miss completely, the shot will penetrate the wall board but much of its killing power will be lost.
Frankly, if an intruder is close enough to grab a shotgun out of your hands, you're no safer with a pistol. He or she (let's not be sexist here and think that all dangerous intruders are male) is too close.
At average self defense ranges that shot pattern isn't as big as you might think, you CAN miss with a shotgun.
I will take your side against MeOnTheBeach on this one.
You have to take many factors in to consideration when choosing a home defense weapon. In the case of your female family member I think Me really did not consider the fact that you were recommending a weapon for a woman.
For a woman or elderly person one has to consider weight and recoil. Many woman and elderly the weight of a fully loader shotgun may make it difficult to control and the recoil of a shotgun may be intolerable.
I will also disagree with Me concerning the intruder grabbing the barrel of the shotgun. Even a shot gun with a short barrel will extend out from you body about 2 feet when at the shoulder ready to fire. A pistol on the other hand can be held at the beltline pointed at a near target essentially not extending from the body at all. This is a shooting technique that should be practiced to ensure that you have learned how to aim in this shooting position.
Another advantage of a pistol is in shooting around corners. With a shotgun you basically have to move in to a hallway exposing yourself. With a pistol you can shoot one handed around a corner or doorway shooing from cover.