General rule when I deal with management : They want paper and numbers. So I give them a decently put together report on the benefits, and an estimate of the win. I add some figures and tables, and keep it maximum 4 pages (more they won't read anyway) with a concise summary on the first half page. You might check the general procedures and office culture at your own work place before going on.
In this case, I'd add :
- a concise description of the concept of SO.
- a table with key programmers on SO, with the amount of answers they give.
- an estimate of how often SO is among the top hits when searching in Google.
- some estimate of the average answering time
- a comparison between the quality of the answers on relevant! questions compared to the ones found in the sources you have available
- if possible, an estimate of the win in time based on the figures mentioned above
A key question I ask myself is : why is it blocked? Is it because a) the decision has been made it is not suitable for work, or b) because everything gets blocked that is not completely approved for work? That's quite an important difference.
If a) it should be not too difficult to convince people that it is suitable for work, given the fact you can easily show a few questions of your own that got answered pretty fast, and show some of the FAQ questions or other more interesting ones that solve important coding problems. Add to this the amount of stackoverflow answers found by simply googling a question, and management will have a tough time defending the decision it is not suitable for work. Tough one in this case is the fact you have to go directly against a decision of management, so somebody might end up with sore toes.
If b), you could actually just check first what the procedure is to get a website approved. I can't see why in this case there should be any trouble in getting SO approved if you follow the right office procedure.