Hi, again it depends what kind of girl she is but Paul's Boutique bags are very popular indeed! I'm early twenties, I have a pink one and love it, and lots of my friends own bags/purses of that brand
www.asos.com/Women/A-To-Z-Of-Brands/Pauls-Boutique-Accessories/Cat/pgecategory.aspx?cid=2704#parentI D=-1&pge=0&pgeSize=20&sort=-1
Some are more expensive than your budget but some are round about the right price.
Harrods also do some cute little bags, if she is one for the names. I also have a couple of these and love them.
www.harrods.com/HarrodsStore/find/c/HOL,harrods-own-totes-and-bags/psi/13
You could maybe get her a couple of different Harrods ones as they are quite a bit cheaper?
Or instead of a bag maybe get her a really cool purse? Like a juicy couture one, they are around £50 and you can buy them off Asos.com, harvey nichols and selfridges have them too.
Again, depends what kind of girl she is. Hope I've helped!