1. She's barely over five feet tall, and she's not very strong physically.
2. As a single mother of two, working full time, her training time would be limited.
3. Budget for the gun, ammo, and accessories would be rather limited. Probably $500 or less for initial purchase, and less expensive practice ammo would be preferred.
4. As alluded to already, young kids are in the house.
Of the 4 listed conditions, the one that always bothers me more than all the others put together is the admission or suspicion that training will be limited by time or budget.
And the rest of the details:
I don't know if she would have the right mindset to use a deadly weapon or the dedication to become properly trained with one. And maybe the challenges that arise from keeping a gun in a house with kids around would outweigh the benefits for her.
In a HD scenario, if your choice of mindset or anything else prevents you from actually using the tool in your hand or nightstand, then the boogieman will have your weapon in his hand, and just maybe it's the only one on the scene, and it is now going from a really bad situation to way way worse.
My suggestion would be to harden the perimeter of her home to the highest degree possible. If her home, that is much easier than a rental, but even as a tenant, many landlords are agreeable that the safety of tenants is a good thing.
If the total budget to increase her safety, somehow, is say, $400, take maybe a third of that and install 1.) better locks on doors and windows, 2.) Motion detecting light fixtures, at front, back and garage doors, pointed out towards the exterior 3.) Become familiar with neighbors and their capabilities and intentions, good or bad. Learn who can be counted on for help when that scary bump-in-the-night occurs. 4.) Develop a plan for home security, regarding answering (or not) the doors and the phone, especially kids that always assume someone at the door is welcome. Keep a cell phone in hand with 911 in speed dial. Lastly, contact local law enforcement to see what response time can be expected, day and night, and be sure to ask them what the history is of your choice of neighborhoods. And if you find your neighborhood is bad for a single mom, figure out how to move to a better one.
A gun may be the worst choice for anyone scared or not trained to use it.