rss245b,
After copying the same project with a different name, You will have to change the context root in the project properties. While applying the changes, it will ask for a confirmation to replace the context root with the new name. Once done, you can see the new name in the url.
Ex : If your project is a web project, then right click on the project > Properties > MyEclipse > Web > Context Root. Modify the Web Context-root field to the new name and proceed.
Let us know how it works for you.