The new ones may be better, but every e-machine I've ever used has had problems, especially the P3 era models. They also tend to use some pretty shoddy supporting parts like PSU's, RAM, etc. I'd recommend a cheapie Dell or a homebuilt rig for her. If she's not a gamer you could build her a tower for around the same as a Dell or emachine even counting the OS. I'd steer you towards Newegg's refurb motherboards section. You can usually get a perfectly good modern Athlon board for around $30. Add another $50 for a midrange AthlonXP, Another $50 to $85 for RAM(256 may be plenty if she isn't doing much with the system), A hard drive on sale for ~$50AR from the hot deals forum, The cheapest video card you can get your hands on(Or just find a nforce2 board n newegg). Grab that lite-On cd burner on sale at newegg and maybe the DVD-Rom as well and you have optical drives. Finally add a cheap case from compgeeks for around $40 or $50.
Grand Total:
$30 motherboard
$50 CPU
$50 RAM
$50 Hard Drive
$35 CD Burner
$30 DVD-ROM
$50 Case + PSU
total: ~$300
+ Copy of XP $200 or better if you get an OEM copy
So you're looking at about $500ish for a minimum-spec tower with OS, adding a monitor will probably get you to about the price of the dell and emachine, but will probably be a better system, with more upgradability for the future and slightly better components(mainly RAM and mobo).