?
In general, we recommend going with a "business context friendly" name for your application. There are several reasons for this, but it mostly goes back to the fact that it's a lot easier to share/consume friendly names.
There are times, however, when it makes sense to match name/code/scm/etc., and this is usually when there's a "conventions" approach. For example, our extensions (see BuildMaster Implementation Specifics) use the application name to determine a lot. But this usually only makes sense when you have a lot of "cookie cutter" componentized applications (such as in our case).
So for your example, I would say name it "Awesome", and then have multiple deployables within that application. I wouldn't use the word "solution" unless it's actually part of the business name. For us, the application name is "BuildMaster" not "BuildMaster Solution"... even though our codebase has a "BuidMasterSolution" folder and a "BuildMasterUbserSolution.sln" file.