You are talking home defense. First, convince her that size doesn't matter. Forget about handguns and look in to carbines.
I personally would look in to the H&K MP5 clones or any semi version of a subgun. The recoil will no longer be an issue and the pistol caliber will be more effective with a longer barrel.
That's good advice and I'll pass that along to her - but I *think* that she's thinking that she may *at some point* in the future carry it (probably would be purse carry). Plus she already has a spring-open lockbox sentry safe kind of deal at her house, for little grubby & curious nephew-hands reasons. Holds handguns but not a carbine.
On the PX4, that too is a good idea - I have a friend who has one of these and likes it. Dang, so many good choices out there - which one is the most economical - probably the XD?
As for carbines, I intend to get her on one of these at some point: M1 Carbine or IMI Timberwolf .357 pump, probably the Timberwolf.
If your girlfriend is new to guns and has no developed skill, have you thought about having her shoot left handed? (or the hand that has no arthritis if there is one). If she is starting at scratch with a handgun it might not be that much of an adjustment.
Good idea; thanks. Except that it won't work, because the arthritis is in both hands - in fact I think it's worse in the left hand.