Wilson has put together some time ago now a great little article on how to add the object picker aka a browse button to the "Send To" page in SharePoint. this has been an asked for feature by many customers and the solution is not that hard. it includes creating a custom copy.aspx, a custom core.js and a custom site definition.
You could also create a feature which adds the custom copy.aspx page to the ECB context menu of an item, but you would end up with two send to options in the context menu.
I prefer the feature approach as it is more resillient to change and service packs. Modifying the core.js means you need to check for changes after each update to make sure that the old core.js syntax still works with the new updated version of sharepoint.