I'd think something with a buttstock would be the easiest to get proficient at quickly. A pistol caliber carbine would get my vote for this situation. Either a lever gun, or - if she can find one - one of the many semi-auto options out there.
Pistols are hard to master, and a lot of smaller statured people find shotguns a chore to shoot.
An AR would also be ideal, but even if you could find one, it would almost certainly be above the given budget. If you find a cache of $500 AR's please PM me
