I had a similar idea a few months ago when I was looking to do some volunteer work. Rather than volunteering at a soup kitchen, or other "unskilled" task I wanted to find an organization where I could volunteer my knowledge of business development, marketing and web dev.
My first thought was to look at websites such as http://www.volunteermatch.org/ . What I found was that the organizations typically requested website redesigns (often using wordpress as the backend) and help with social media management. The time commitment of these opportunities were as you stated above, longer term requiring a few months. The scope of work provided was often inadequate and would require you to contact the organization for details.
Solution:
Create a guide on how to assess requirements and create a proper scope of work for projects, the organization seeking volunteers would submit this as part of their application[1]. Then have a volunteer that would make an appointment to speak with the organization and refine this SOW into a workable project.
Once the project was accepted and requirements laid out; open the opportunity for contributors to offer their time and create a core team to start the project. For website redesign/management use a common codebase that could be reused for projects which would reduce the on-boarding process. This would also allow general documentation and training materials to be created so the management of the website could be transitioned to a full-time volunteer/employee of the organization.
Once the common core was developed and a repo created, issues/tasks would be created and open for contribution which would give devs willing to volunteer a few hours some opportunities to contribute.
I think a handful of well written guides for marketing, social media management, front-end/backend documentation and possibly even basic web design tutorials would go a long way towards improving organizations. A second level of this would be to have recurring roundtable meetings (google hangouts?) where more general individual problems could be addressed allowing for the main project scopes to be slimmed down.
1. Also, works as a weeding out process by requiring a deeper commitment. Organizations would have to to follow to guide, watch a video and think through their needs as opposed to posting a paragraph asking for help.
Related experienced dev/intern pairing program: A non-profit named Launch Code (http://launchcode.us/) was recently started by Jim McKelvey (co-founder of Square) that places new developers with companies for paid pair-programming internship.