Welcome to Android Central! From a techie standpoint, the Nexus 5 is probably the better phone. Despite it being almost 2 years old, it is still very fast, has an excellent screen, and has the advantage of being a Nexus device--so it should receive Android M when it comes out (although that'll probably be the end of the line for official system updates). I find the camera to be quite good as well. The main downside is very mediocre battery, so if she's a heavy user, she'd have to charge up at least once in the middle of the day, maybe twice.
The Desire Eye's claim to fame is selfies--both rear and front facing cameras are 13 MP, so the image quality of the selfies will be high. It also has an SD slot, which is important if she plans to store lots of music, photos, and videos locally. I'm not sure what the battery life is like, but the 2400 mAh battery is only a bit bigger than the Nexus 5's 2300 mAh battery, so it might not be much better. So if selfies and using the phone as a media machine are more important, then the Eye might be better for her. It's unlikely that the Eye will get more system updates--I think it got up to 5.0.2.