My first response is a Webkinz (stuffed animal). They run about that (usually ~$14 for a big one and ~$11 for a small one) , and are universally beloved by 6-year-old girls.
HOWEVER, they require a computer to really get the most out of them (the stuffed animal comes with a code that allows you to play with a virtual counterpart) and "a family in need" probably won't have a computer.
Sozlet got a small electronic game from "Littlest Pet Shop" for her 6th birthday that she loves and was probably about $15, but I can't find anything similar on Amazon.
Groovy Girls are about $15 -- six years old was about the peak of Sozlet's interest in them:
They always have a holiday-themed doll, which would be perfect.