Could you elaborate a bit more what she'll be using it for? There's a lot of other factors when choosing a somewhat future proof phone (if you don't plan on changing it within 2 years) - screen size, camera, onboard memory, ram, the list goes on. If you won't be getting an upgrade within that time interval I'd go for something with a 5in + screen and more than a gig of ram. Plus perhaps 16gb of storage instead of 8 if you won't be using an SD card.
My mom only uses her phone for facebook, email, calling and the occasional youtube video, and any phone can handle that, even the cheapo samsungs that are stuck on android 4.1~4.2. I'm saying that cause she's on a 1st gen Galaxy Core and there are barely any hickups at all. If your mom will be doing anything more demanding than that I'd go for some of the devices on the second list. Yes, the Xperia E4 is alright, but you can get more bang for your buck with some of Samsung and Sony's midrangers or HTC's older Desire phones for example.
As a user of HTC I can tell you their flagships suck, but the value oriented Desires are pretty great. Look for the Desire 816 and Desire 626, which are both well below $200, in some places below $150 and are pretty adequate fairly premium phones. You could probably even find a One M7 for around that price if you look hard enough and that's an iconic 2013 flagship. What I'm trying to say here is that cheap entry level crapphones really aren't worth it when you have decent midrangers for as little as $20 more. So do some research. If you're in the states you can get a nice carrier dieal with AT&T or Verizon on the phones I mentioned above. Good luck and hope I helped