Well, not sure this is the cheapest computer ever, but she'll love it -- the case is little and it's red
. Girls love red. Plus, I've actually built this PC (in black) and it kicks serious azz. It is 100% high quality parts from the best manufacturers -- no corners cut.
http://lian-li.com/v2/tw/product/upload/image/Q7/pc-q7-20s.jpg
LIAN LI PC-Q07R Red Aluminum Mini-ITX Tower Computer Case
http://www.newegg.com/Product/Product.aspx?Item=N82E16811112246
$59.99
Intel Celeron E3300 Wolfdale 2.5GHz LGA 775 65W Dual-Core Processor BX80571E3300
http://www.newegg.com/Product/Product.aspx?Item=N82E16819116264
$51.99
Intel BOXDG41AN LGA 775 Intel G41 Mini ITX Intel Motherboard
http://www.newegg.com/Product/Product.aspx?Item=N82E16813121441
$74.99
Kingston ValueRAM 2GB (2 x 1GB) DDR3 1066 (PC3 8500) CL7 KVR1066D3K2/2GR
http://www.newegg.com/Product/Product.aspx?Item=N82E16820134492
$38.99
SeaSonic SS-300ET Bronze 300W ATX12V V2.3 80 PLUS BRONZE Power Supply
http://www.newegg.com/Product/Product.aspx?Item=N82E16817151086
$39.99
Western Digital Caviar Blue WD3200AAJS 320GB 7200 RPM HDD
http://www.newegg.com/Product/Product.aspx?Item=N82E16822136098
$44.99
LITE-ON CD/DVD Burner SATA iHAS124-04
http://www.newegg.com/Product/Product.aspx?Item=N82E16827106289
$16.99
Total:
$327.93 w/o tax & shipping
WEI on the CPU was 5.9 and the graphics 4.3. It runs Win7 like a champ. It'll definitely work for internet, music, surfing and even some movies. If HD movies are necessary you can move up to this mobo for $25 more:
Intel BOXDG45FC LGA 775 Intel G45 Mini ITX Intel Motherboard
http://www.newegg.com/Product/Product.aspx?Item=N82E16813121353
$109.99
but then you'll need to go with DDR2 800 memory not the DDR3 above.