Are there any restrictions/preferences on size or looks?
The biggest problem is going to be the ".1" - you just can't expect a truly good subwoofer when you're looking to get 5 speakers PLUS a sub, all for under $300. That said, so long as expectations are held to a reasonable level, you can find some ok "entry-level" 5.1 packages for that price.
My favorite, small and darn good sounding 5.1 speaker package was the Athena Micra6. Sadly, Klipsch pretty much killed the Athena brand when they bought them, but you can still find the Micra6 package used or refurbished.
Athena Micra6 at Amazon
One of the very best "bang-for-buck" inexpensive speaker systems is Onkyo's SKS-HT540 7.1 package. Your sister might not need the extra two speakers, but for the price, I'm sure she can use them in another room or just keep them as spares or even sell them
Do not be fooled, these speakers are quite large. A lot of people are surprised by their physical size, so if she want really small and "stylish" speakers, then this Onkyo package isn't for her. If whe cares more about sound quality than looks though, this is one of the very few inexpensive speaker packages that doesn't try to get around basic physics with nothing more than marketing BS! These are honest to goodness speakers that just happen to be very inexpensive.
Onkyo SKS-HT540 7.1 package at Tiger Direct
Those would be my choices in this price range
