In a plain ol ASP.NET application, the screen refresh can be eliminated by using update panels in webforms, or by using more of the AJAX capabilities of the MVC Framework. (Wikiplex on Codeplex has a crazy simple example of this that I love in their example application).
Between Silverlight and Flash, however, Silverlight is designed to be a Web Application, or (at times) a desktop application in a Browser, while Flash is more of a multimedia framework.
Since it is school administration, which doesn't require random people to have Silverlight installed, Silverlight and JS/AJAX are both good options. Both are built into the ASP.NET set of libraries, and have good support in Visual Studio. Flash is a bit of an outsider in this regard.
So to sum up:
In general JavaScript/AJAX > Silverlight but either would be great.
If you are totally against any screen refresh, I'd lean more towards Silverlight.
If the developers are very familiar with Flash and communicating with the server with Flash, then I'd say it is also an option, but with no knowledge on all three and I'd not pick Flash first.