I am an avid amateur photographer, and this is what I found over the past few cameras that I have had. At $300, you are looking at a point-and shoot camera if you choose new. There are many good options out there, but most will not be an upgrade to your phone. They are harder and less convenient to use than a phone, and the lack of control over settings will be frustrating. Most of all, you don't "feel" like a photographer, you feel like Aunt Marge with her old pocket camera. Most people just revert back to their phone.
My first recommendation, therefore, is to look for a used, compact mirrorless camera.
My first was the Nikon J1 and I cannot recommend it highly enough for anyone looking to make a move up from a cellphone camera. I still have mine as a backup because I like it so much. It is small, has interchangeable lenses, a built-in flash and has plenty of accessories available. It is super easy and convenient to use, and works great in low light and for stopping fast action. What this means is that she will be amazed at how good her pictures look, and she will feel like a real photographer. You can get a full kit on eBay under your budget easily. Nikon made a few other models (V1) if you want a hotshoe for a flash or even a completely waterproof version.
I preferred the Nikon over other models because it had the best picture quality for beginners. Other models like the Sony and Panasonic could produce better picture quality if you play with settings and know what you are doing. But if you want to turn it on, point, and shoot at your kid's recital or soccer game or whatever, you won't beat the Nikon. That said, there are some other great options out there like the Sony Alpha, Olympus PEN series and Panasonic LUMIX series. All of these produce amazing results compared to a phone, give the user plenty of controls to play with, but are still small and convenient to carry and travel with.
The next option is a DSLR look-alike, usually a super-zoom like the Nikon Coolpix.
I had one, and they have the look and feel of a DSLR, and they are very convenient and have a long zoom range. They are a great upgrade if you don't want to have to deal with carrying an extra lens for zoom work. I found that that the picture quality was just not as good however. I was never able to get the sharpness and detail that I got with my J1. I eventually sold mine.
The third option is a package deal or a used DSLR. Every year, Costco for example has a package deal on a DSLR with 2 lenses and a bag. Sometimes they can be close to $300, other years the are more. You might get lucky and get a cheap deal this year. If not, Wal Mart and Target always have an entry-level DSLR for around $300. This is this year's from Wal Mart, the Canon EOS Rebel for $297:
You can also find packages like this on CL and EBay all day long. People buy them and never use them, or move up to a bigger better setup, and the old ones go for a good price. Don't be afraid to negotiate, people often list them for what they would like to get, but the market is actually a lot lower. Also check out the pre-owned section at B&H and Adorama. I got my current set up from their pre-owned section and you can't tell it from new. They actually list detailed information about the condition and it is accurate.
If you think your budding photographer will take it seriously for a while, this is absolutely the route you want to take. There's just nothing like the quality of a DSLR. People have their preference for Nikon, Canon etc but for a beginner photographer each will represent a major leap in quality. If there are going to continue with this hobby, they may want to add lenses and accessories, so I would avoid lens system that is not as common as the big ones.