I'm a budget gun guy myself, so I understand the mindset. I started with cheaper guns and gradually moved my way up. Probably spent more in the long run, but gained some experience along the way.
For the purposes of home defense, as others have mentioned a shotgun would be a good idea. Add a light to it with some low recoil buckshot and it can handle most home defense situations, especially living alone. In a home defense situation, your best bet is to call 911 and hunker down in a safe(r) room and wait for help to get there. Clearing the house is unnecessary and more dangerous in most cases, and really you should only do it if you have to. For that kind of situation, a shotgun is key. If she is inexperienced, a 20 gauge shotgun with and HD barrel or a "bantam" design could be a very cost effective option.
If a shotgun is too intimidating or just not the direction she wants to go, a pistol caliber carbine could be an option as well. Case in point, the Hi Point Carbine. Worth consideration especially since it usually can be had under $300. Easier to shot than a handgun and with a laser or red dot it can be relatively easy to shoot for a novice. My one big qualm with it is how difficult it is to break down and clean.
Also, if she is under 21, long guns like above would be her only legal option, figured it's worth mentioning since her age was not clarified.
If she really wants a handgun, these are my recommendations in each price range. I would personally stick to 9mm, .380, or 38 special to keep ammo cost and recoil down, but YMMV. I'm going to focus on the $200-300 range, as under $200 a Hi Point C-9 is about the only thing I MIGHT consider in a pinch. At $300-$400, way too many options to list them all, but for $300-$400 you have a smattering of offerings from S&W, Ruger, CZ, Kahr, Glock (if you can get a good deal) Springfield, etc.. if you shop around and shop smart. When you get into this range, you aren't really considering "budget" options. Those at the lower end may be the budget models of their lines (SR9E, SD9VE, etc) but are still high quality guns from well known manufacturers. You also have Canik in the lower end of that range, who are becoming a more common name.
Finally, these recommendations are having no idea whether concealed carry is a consideration or not, so threw subcompacts/compacts/midsize/full size all together. So, anyway
$200-$300
Taurus PT111 & 709 - Great guns for the price. Lifetime warranty, but customer service can be slow and not the greatest. You can get these under $200 sometimes, but they are more commonly found in this range.
Armscor M200/M206 - The 200 is the 4 inch barrel, 206 is the snub nose 2 inch barrel. They aren't pretty and can be rough around the edges, but they go bang reliably. If someone wants an inexpensive revolver, it's hard to go wrong with these. Armscor's customer service is top notch as well.
SCCY CPX1/CPX2 - American made and an unbelievable warranty. Trigger pull is long for sure, which is why I personally don't like them but I'm spoiled. My uncle has one and absolutely loves it. Good value.
EAA B6P/B6P Compact/Pavona/K2P/Witness P - EAA imports quite a few CZ-75 clones in this price range. They don't share parts with the CZ-75 to my knowledge, but are based on a sound design and are very reliable. My first handgun was a B6P and it is what got me into CZ's. IMO, these are one of the best values for your dollar.
Zastava M88A/M70A - These are rougher, Com-bloc guns that probably wouldn't appeal to her as much but are reliable home defense options, or collectible for people who like Com-bloc guns.
Bersa Thunder 380 - Heard good things about this gun, but admittedly have never shot one myself. I personally would have the extra power of 9mm over .380, but alot of people really like this one.
Walther PPX - It's ugly IMO, but a good value in the price range. Great capacity and well built.