Everything a church does increases the workload of someone in some way, but to answer your question, all 'free' mail accounts either use your data to enhance search results or generate ad revenue. Just one thing that comes to mind is offering members ads that come from a Christian worldview. For instance, information on global missions, evangelistic outreach and volunteer opportunities.
Providing a permanent address for mailing can be a way to keep in touch with individuals who may have a host of issues. In dealing with the mentally ill poor, many choose to live independent lives, but connecting with them can provide a bridge to family members and public health professionals. It can also provide a way to ensure they eat and are keeping up with prescribed meds. Really, I'm thinking about relationships though.