This is interesting! I was not aware that local MPs can do something about individual migration visas. How do they assist with the process?
If there are compelling/ compassionate circumstances,intractable technical problems or obfuscation on the part of the Department, a good MP can contact the Department or use his internal contacts to resolve matters.
If the application is within standard processing times and there is no particular reason for urgency, they will usually not intervene.
My local MP is very active and supportive in deserving migration cases and has assisted me several times with Ministerial Intervention requests. He also assists with employer sponsorship issues and the like.
So don't be afraid to call your local MP's office and ask for assistance when it is warranted. Be nice, be polite and be well prepared with relevant documentation and a realistic case for assistance.
According to my local MP, most of the issues they have to deal with on behalf of local constituents relate to Immigration and Social Services.