What motherboard does your daughter's PC have? Does it have a PCI-e 1.0/1.1 or PCI-e2.0 slot?
The older PCI-e 1.0/1.1 might have some compatibility issues with a PCI-e 2.1 card. Rare, but it has been reported. Therefore, a Radeon HD 6870 has a small chance of not working. Anywaste, I say the HD 6870 is a bit overkill for 1280x720 resolution and it is also consume more power than the HD 4850; 156w vs 146w at maximum load. The nVidia 560 Ti is even worse at 218w; it's also overkill for 720p.
I recommend a Radeon HD 5770 or HD 6770; both are really just the same card with different names with a maximum power consumption of 108w. It is also a little more powerful than the HD 4850. You can probably find one for around $80.
If your motherboard has the newer PCI-e 2.0 slot, then I recommend the newer Radeon HD 7750 which is a PCI-e 3.0 card. Technically it should be backwards compatible with a PCI-e 1.0/1.1 slot, but it is possible there could be issues. It's maximum power consumption is 46w and it is overall roughly equal to the HD 5770/HD 6770. The HD 7750 sells for $110.