No need for a video card, that's the whole point of the APU line anyway. I adjusted a couple of things in your build and added things you left out.
1st, unless your budget is incredibly thin, for the love of god don't go with a refurb hard drive. Its not worth the $20 for the headaches you're opening yourself up to.
2nd, for wifi, go for a board with onboard Wifi. It's going to be cheaper than buying a sub par addon card or dongle. I adjusted your build to have one. If your wifi coverage sucks, look at powerline adapters. If she's going to be doing it all online, it does no good to have no connectivity.
3rd, make sure you get a good PSU. You don't need a lot of power, but you are going to want either Semi or Full modular. That case has very little clearance for the massive bundle of cables that will come with a non modular system and it'll heavily impact any airflow in an already cramped case. Also, and this comes from someone who deals with this stuff on a regular basis, it will make your life with this computer so very much easier. If you can't get the one I listed, here is a good forum resource for PSU's to look at that are quality parts. While your build isn't a high power draw, you still want quality parts for this of all parts. Any build with a crap PSU will eventually have more problems than it's worth.
www.tomshardware.co.uk/forum/id-2547993/psu-tier-list.html
4th, you need an OS. Just go for Win10. Its the cheapest and MS is making sure everyone swallows the Kool-Aid anyway. Plus it'll mean she'll be compatible with any software coming out that she'll need at home.
5th, consider upgrading the HD to a 1TB, its like $5 more for double the storage. Thats a minor and personal nitpic tho so take it as you will.
6th, and finally, keep an eye on temps. The AMD APU line has a low heat ceiling and you're in a small case. Test it out with the proper software and if you're spiking too hard, consider an after market cooler like the Cryorig M9a. Its short enough for the clearance listed in the case and will be much quieter than the stock fan. Keep in mind, understanding temps for AMD cpu/apu is different than Intel chips. Read up on it here
http://www.tomshardware.com/faq/id-2122665/understanding-temperature-amd-cpus-apus.html
PCPartPicker part list /
Price breakdown by merchant
CPU: AMD A6-5400K 3.6GHz Dual-Core Processor ($39.88 @ OutletPC)
Motherboard: Gigabyte GA-F2A88XN-WIFI Mini ITX FM2+ Motherboard ($92.99 @ Directron)
Memory: Corsair 4GB (1 x 4GB) DDR3-1600 Memory ($17.98 @ Newegg)
Storage: Western Digital Caviar Blue 500GB 3.5" 7200RPM Internal Hard Drive ($41.98 @ Amazon)
Case: Cooler Master Elite 130 Mini ITX Tower Case ($42.99 @ SuperBiiz)
Power Supply: SeaSonic 520W 80+ Bronze Certified Fully-Modular ATX Power Supply ($65.98 @ B&H)
Optical Drive: Asus DRW-24B1ST/BLK/B/AS DVD/CD Writer ($18.89 @ OutletPC)
Operating System: Microsoft Windows 10 Home OEM (64-bit) ($84.99 @ NCIX US)
Total: $405.68
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2016-05-09 06:55 EDT-0400
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