Beth,
I think it will depend a great deal on what specifically you want to accomplish... there are quite a few open-source titles out there that could be used in a hosted environment, with your primary interface being a browser. Almost everything I can think of is going to be biased, because I like LAMP (Linux, Apache, MySQL, PHP) a lot.
Drupal.org comes to mind right away. (www.drupal.org)
Another really great place to look for similar open-source software is http://freshmeat.net
Principally, each of the systems you find will be geared to this function, that function, etc. But I feel confident that an appropriate use case analysis would provide the additional information required to "extend" the database on any of these products and move forward. (WOrk smarter, not harder, etc.)
What's your time line? I've been graced with a google wave account in the current sandbox, but it won't be available for the general public until next year. I have a strong feeling Wave is going to turn a lot of what we do with the Internet on it's ear, and I can see non-profits making HEAVY use of it if it makes good on it's current promises.
Dustin Decker
1 On 1 Security, Inc.