First, yuck! Windows 10 is freaking $120 for Home? When are they going to jump on the $50 bandwagon?
Now to the PC question. Assuming around $400 left, I'd say you're looking at a VERY barebones PC. I mean, it'll work and be functional, but it won't FEEL like a gaming PC that you'd recognize. It'll feel like a kids gaming computer. I did roughly this price point for a PC for my kids (7 and 9), minus quite a bit of dough because of some leftover parts, and it feels cheap. I mean, it's fine, sure. But I wouldn't recommend it for a teen who needs to know the burgeoning love of the PC gaming hobby. :) Another $200 or $300 (I know, I know) and you really start to feel like a first-class citizen though. I usually put the sweet spot of price/performance around $750, but maybe that's just because I have some mental model of that being the right price. Let's look at builds.
Two resources which help answer your question:
This slick website (which sadly doesn't get a cool preview box):
And then NeoGAF's build a PC thread. The google doc link is dead but the embed is still there.
General recommendations at the ~$700 range (hopefully also apply at ~$400):
1) For GPU, go with AMD (480 4GB or 470, or 460 for super cheap). Don't do the 3GB Nvidia cards. FreeSync monitors are a cheap future upgrade. G-Sync is expensive. AMD is good at the low end.
2) Favor 16GB of RAM, even if you need to cut somewhere else. 8GB really won't feel like enough unless she's only doing one thing at a time. 20 chrome tabs and a game and 8GB will choke. "Well, then don't do that!" doesn't work with a teen. Teens are busy multi-taskers and you can expect a lot of things happening.
3) Spend some money on an SSD, even if it's only 120GB. Pair it with a larger magnetic disk and put the OS on the SSD. Or just go with a 256GB SSD and add a HD as a birthday present later. Most everything is online now and you can share a folder with her to grant her disk space if she needs more.
4) You don't need to overclock the CPU. "But it's so easy with Intel!" So what. The locked CPUs are cheaper and plenty fast.
5) You don't need a 750W PSU. What is with people these days? I had a 450W Thermaltake survive 4 or 5 different builds with various hardware, including power-hungry AMD CPUs along with power-hungry AMD video cards. It's fine. Budget the power you need and then exceed it just a little.
Edit: Rearranged the links because NeoGAF is a little outdated. To highlight: http://www.logicalincrements.com/
Edit 2: This is the best site for building, suggesting, and sharing PC builds: https://pcpartpicker.com/
Edit 3: Apparently I forgot that pcpartpicker also allows for sharing and ranking builds, so from their list of builds I found this one for $550 right near the top: https://pcpartpicker.com/guide/jnFfrH/entry-level-gaming-build
With pcpartpicker, add some money if you don't want to buy from like 5 different stores to get the cheapest price. :/