Every group should exist for a purpose beyond meeting together. At Calvary, we talk about groups being "missional" because we want them to have a mission they are accomplishing together. I think this is a healthy endeavor for any group.
If your group is looking for an easy first step into having a missional mindset, I'd recommend KIVA. This is nothing more than a financial investment (no time, no effort), which is often the easiest way for people to begin their "generosity journey".
I'm a big fan of Kiva, and have invested with them on several occasions. Using Kiva, myself and a many other people have banded together to loan people the money they need to launch their own businesses (like buying a cow and selling its milk). The'll pay us back eventually and with my Kiva account, I'll use the money I get back to help make another loan to someone else.
It's certainly sobering to think that for less money than it costs for my family to eat a great dinner someone can participate in a project that can radically alter the course of another family's life. Generations of these people's family might be better off because I participated.
Your group could probably give more. If they were willing to pinch a little, they might even make the entire loan themselves. But for now, begin by taking baby steps toward a MORE GENEROUS life.