I agree that volunteering is a good route. And I agree that it should be a program that you select together...and do together. Rather than her having a child to mentor, I'd suggest that you, and other adults, mentor her first. Your modeling of compassion and community service is so important for her to experience and you are her best teacher. Developmentally, she may not be ready to mentor another child, particularly a child from another culture or community, until she has experience and some concrete tools, which she will pick up alongside adults working at a PADS program, cleaning toys at a shelter, planning and running a charity garage sale, collecting books to donate to the Infant Welfare Society, doing yard work for an elderly neighbor each week, etc.
In terms of always wanting this or that, most kids eventually develop the ability to delay gratification. Some kids do it earlier and more fully than others. But practice and again, modeling, is a good thing. So make it a point to say things like, "Yes, that movie looks great. Let's wait two weeks and see if we still want to go!" Or, "I really want to read this book. I will have to stop by the library and order it" (rather than picking it up at Borders).
When she requests an item that a friend has you could say, "That sounds interesting, but that is not something that our family is able (or willing) to buy." And explain why (insert desired item) doesn't fit your values or budget. Could she borrow the item from the friend, if it is appropriate and something you don't object to? And loan the friend something she enjoys.
Also, we have a rule in our house that if something comes in, something has to go out (donate). So, if we pick up some clothes for the kids (and we actually get a lot of hand-me-downs and items through www.Freecycle.org) then the children go through their clothes and select items to donate to a shelter. Same goes for the adults.
This might be pushing it, but how about for her birthday she collect new books to donate to a children's hospital or clinic for children with special needs, rather than receiving presents? You and she could research options and deliver the books together! (I have leads if you need them!)
Good luck!!