An NRA-certified instructor is your most likely bet.
You can find one in your area here:
http://www.nrahq.org/education/training/index.aspLook specifically for classes in Home Firearm Safety, or FIRST Steps Pistol Orientation to start. A decent instructor will provide a .22-caliber pistol and ammunition, so you (and/or your mother) can get to grips with the basics of firearm handling and marksmanship before you commit to buying a firearm.
If you, and your mother, do decide to go that road, you can next look into a Basic Personal Protection In The Home class. You will probably have to provide your own firearm and ammunition, so you can train with the weapon you actually intend to use. I can't tell you what firearm to get, because there's no "perfect" handgun; personal preference simply plays too large a role. See if there's an indoor range that does rentals and try out some models. That said, there are two options that I always advise prospective handgun owners to look at first, because they've worked well for a large number of people.
If your (mother's) preference goes toward a revolver, I'd suggest a Ruger SP101 with a 3" barrel, chambered for .357 Magnum. Despite the different numbers, the .357 Mag cartridge is essentially an elongated and more powerful .38 Special, and you can safely load and fire .38 Spl cartridges in a revolver chambered for .357 Mag. And because a revolver chambered for .357 Mag is designed to handle the higher pressure and recoil of that cartridge, its mass absorbs a lot of the recoil from the less powerful .38 Spl (even in "+P" higher pressure loads), making to quite comfortable to shoot. So get a revolver chambered for .357 Mag, but load it with .38 Spl; you can move up to .38 Spl +P or even .357 Magnum once you get comfortable with it.
For semi-automatics, the first thing to look at is the Glock 19. Glocks are really easy to operate and maintain, as well as being reliable and accurate, which is why they're popular. The "compact" frame of the models 19, 23 and 32 are a comfortable fit for most hands, while the 9mm chambering of the 19 provides adequate incapacitating power with a level of recoil that's manageable for most people.