It makes a big difference if she's doing sport/rec riding or commuting/errands type of thing.
In either case:
- gloves, helmet, padded shorts
- glasses w/multiple lenses (clear & smoke)
- water bottle & rack (2 for long rides)
Riding for fun/exercise:
- seat pack or handlebar bag
- some way to stow a jacket - rack and bungees?
Riding as transportation:
- good lock (I have a modern u-lock with a cable)
- front & back lights
- rack & bungees, maybe a rack trunk w/shoulder strap or basket/pannier system
- fenders
- messenger bag with sternum strap (somehow they're more comfortable than backpacks but YMMV)
I would say a tube and tire levers, but I've actually never had a flat when riding (okay, once, but I noticed it 50 yards from the car on my way back), so I can't vouch for that. Obviously she needs a good pump at home, I like the ones with the gauge built in. Supposedly you can use a dollar as a temporary patch if you have tire levers and a pump.
For the cycle computer, if she has a smartphone she might get almost as much utility out of a GPS-based phone app that maps your route and tracks speed, etc. (But I keep buying computers and never getting them installed properly, so I don't really know what I'm missing.)
Finally, it sounds like she's really into the clipless shoes, but I never got the hang of that. I love, love, love my PowerGrips (http://www.rei.com/product/788129 ), you get good pedal contact, don't have to worry much about clipping in or out, and can wear any type of shoes. And they're way cheap.
For online shopping, Nashbar and Performance are the biggies but their low-quality stuff is pretty low-quality. I also like Price Point and REI-Outlet. Terry has an online store (premium quality women-specific gear at a premium price) but you can also find their stuff at Team Estrogen and occasionally at Campmor or Sierra Trading Post (all reasonable online vendors).
Yes, I'm an expert shopper. If I spent half the time doing my sports as I did shopping for the gear I'd be mighty!