I'm assuming she has made the decision - that she thinks she needs some kind of defensive firearm at home....so ....a couple of things...
A shotgun may not be her best choice. Recoil on any fixed breech shotgun ...even in a 20ga ...may be too much for her. But if she has decided she wants a shotgun / given that she is so petite...a semi-auto shotgun ( so the action absorbs some of the recoil ) ...like a Remington 1100 or 11-87 / even used...is a much better choice than a pump gun.
A handgun may be a better choice...but recoil / what fits her hands the best, etc ...is a big deal as well. For a more petite person / with smaller hands...something in a .380 or 9mm is probably about as big as you want to go ...and maybe something in a single stack / like a Sig 239 in 9mm is a good option...or a 1911 in 9mm ...( again, slim profile). Most double stack handguns will be too thick / and not comfortable for her to shoot ...and don't go any bigger than a 9mm
I have some ladies in my family that shoot occasionally ....even some young granddaughters....and 9mm is fine ( Sig 239's, Kimber tactical pro II model in 9mm, and maybe a shotgun in 28 ga ....20ga is big stretch / unless there is a lot of upper body strength ...and that varies a lot by individuals.
I've had adult women that couldn't handle the weight and recoil of a 28ga ...and yet some of my 13 yr old grandaughters can ....so its a hard thing to call / or recommend...