I like the .22 Win mag. Mine's a Taurus 941 (8-shot) DA revolver. There's no recoil so your friend will probably like it. It does have a heavy DA trigger though. Inexperienced shooters may not like it. But you can always get some trigger work done.
Another option would be for a .32 ACP semi-auto. This is another round I really like due to low recoil. I have a Walther PPK/s (one of my favorite shooters) and a Beretta Tomcat. The Tomcat makes a great purse gun because of it's size, and the fact that it has a tilt-up barrel. Your friend wouldn't have to rack the slide to load it, a bonus for those with arthritis. There's a lot of different .32 semi-autos out there, look around for the right one.
Another option of course is to have your friend practice practice and practice some more on the .380 until she gets used to the recoil.
I have rounds yet to fire before I sleep ...