While the documents are correctly saved with the correct new name (e.g. Map1.dwg, Map2.dwg, etc.) in the correct location, the doc.Name properties are not updated, the title bar of autocad also. They still remains Drawing1.dwg, Drawing2.dwg, etc.
I have used the dynamic keyword available in .Net framework 4.0. If you are using AutoCAD 2011 which targets the 3.5 framework, you can replace it with AcadDocument and add references to the AutoCAD COM interop assemblies.