OP: your daughter's computer has a 5 year old processor which wasn't great at the time (review
), but it is a serviceable desktop still. Particularly with 8 GB RAM, and I guess Windows 10 (though maybe still Windows 8.1) and you might be able to get another few years out of it before it becomes sluggish, maybe even another 5 of usage too, if you're lucky.
However, you can't exactly expect it to last, the intended lifespan of these things is 3-5 years. If it does, great, but I would be cautious about putting money into it at this time, and probably wouldn't, especially if budget is tight. Now would be a good time to start saving up to replace the whole machine. If she's going to be old enough to have some independence in a couple of years, I might start saving for replacing it with a laptop, maybe starting that off with some of this birthday's budget if that's the sort of thing you could do?
However, if she's gaming and her PC really can't keep up, I would consider a graphics upgrade, to keep her happily playing and enjoying
the activity and the happiness it can bring, rather than be frustrated by the limitations of her PC. You do have to be very careful on power budget, though, because that computer's power supply is not so great. Probably a ~50 W card is okay, I wouldn't want to go above that really, and I think you absolutely have to stay under 75 W.
I would look at a $130-$150 AMD Radeon RX 560 4 GB, the 14 CU (896 SP) version because it uses less power than the 16 CU (1024 SP version) (example
). But this is only if she is running Windows 10. If she is still on Windows 8.1, it will need to be a NVIDIA graphics card due to lack of Windows 8.1 driver support from AMD.
The only low-power usage NVIDIA card in the current generation is the $90-$100 GeForce GT 1030 2 GB, which I don't recommend despite owning one. It's too slow to be worth spending money on, and has too little RAM too. Even Rocket League on that card is a struggle. It will be probably better than the integrated graphics which the PC has, but not by much. The GT 1030 2 GB which I have is not better than the Radeon HD 7750 1 GB previously installed, and your daughter's Radeon HD 8670D is probably not too much worse than my old HD 7750 (though probably a gerbil here will correct me...)
But again, I would buy it in your shoes, if my daughter was very frustrated with her gaming hobby by the PC, and if I had no way to provide a better / replacement PC within the next few months -- keeping the happiness from the hobby is so important with kids...
HTH, and good luck with the birthday present whatever you decide to do!