Not a build-your-own, but if all she does literally is surf the web, you could consider a Chromebox and save yourself $100.
http://www.newegg.com/Product/Product.aspx?Item=N82E16883220749&cm_re=chromebox-_-83-220-749-_-Product
For build-your-own, the lowest I can get, including Windows, is $285. One advantage, assuming she is not storing videos or anything, is this build would get her a cheap SSD.
PCPartPicker part list:
http://pcpartpicker.com/p/dPncP6
Price breakdown by merchant: http://pcpartpicker.com/p/dPncP6/by_merchant/
CPU: Intel Celeron G1620 2.7GHz Dual-Core Processor ($36.99 @ SuperBiiz)
Motherboard: Biostar H61MGV3 Micro ATX LGA1155 Motherboard ($34.98 @ SuperBiiz)
Memory: Crucial Ballistix Sport 4GB (1 x 4GB) DDR3-1600 Memory ($28.90 @ Amazon)
Storage: Kingston SSDNow V300 Series 60GB 2.5" Solid State Drive ($45.99 @ Amazon)
Case: HEC Vigilance MicroATX Mini Tower Case ($23.99 @ SuperBiiz)
Power Supply: Antec 450W ATX Power Supply ($24.99 @ NCIX US)
Operating System: Microsoft Windows 7 Home Premium SP1 (OEM) (64-bit) ($88.98 @ OutletPC)
Total: $284.82
Prices include shipping, taxes, and discounts when available
Generated by PCPartPicker 2015-03-16 14:40 EDT-0400