Not sure if this is even possible. Is there a way to setup a global property that can be used by all webparts on a page.
If you are talking about a Publishing page by any chance, you could use any of the fields of the underlying Content Type via the Page Field filter web part and send that to any of the web part in the page via connections. Of course not every web part support Connections, but most ListViews or DataView support that.
Additionally you could simply use any of the Filter Web Parts based on existing columns in lists of your site, and again use Connections to send values.