I developed a dynamic web project in Eclipse. I can access the app through my browser using the following URL:
http://localhost:8080/MyDynamicWebApp
After changing the context root in project properties you have to remove your web application from Tomcat (using Add and Remove... on the context menu of the server), redeploy, then re-add your application and redeploy. It worked for me.
If you are struck you have another choice: select the Tomcat server in the Servers view. Double clicking on that server (or selecting Open in the context menu) brings a multipage editor where there is a Modules page. Here you can change the root context of your module (called Path on this page).