From a design perspective, I think it is wrong to require a static implementation member... The relative deference between performance and memory usage between static and not for the example string is minimal. That aside, I understand that in implementation the object in question could have a significantly larger foot print...
The essential problem is that by trying to setup a model to support static implementation members that are avaialble at a base or interface level with C# is that our options are limited... Only properties and methods are available at the interface level.
The next design challenge is whether the code will be base or implementation specific. With implementation your model will get some valdiation at compile time at the code of having to include similar logic in all implementations. With base your valdiation will occur at run time but logic would be centralized in one place. Unfortunately, the given example is the perfect show case for implemntation specific code as there is no logic associated with the data.
So for sake of the example, lets assume there is some actual logic associated with the data and that it is extensive nad/or complex enough to provide a showcase for base classing. Setting aside whether the base class logic uses any impelementation details or not, we have the problem of insuring implemtation static initialization. I would recommend using an protected abstract in the base class to force all implementations to created the needed static data that would be valdated at compile time. All IDE's I work with make this very quick any easy. For Visual Studio it only takes a few mouse clicks and then just changing the return value essentially.
Circling back to the very specific nature of the question and ignoring many of the other design problems... If you really must keep this entire to the nature of static data and still enforce it thru the nature confines of the problem... Definately go with a method over properties, as there are way to many side effects to make go use of properties. Use a static member on the base class and use a static constructor on the implementations to set the name. Now keep in mind that you have to valdiate the name at run-time and not compile time. Basically the GetName method on the base class needs to handle what happens when an implementation does not set it's name. It could throw an exception making it brutally apparent that something is worng with an implementation that was hopefulyl cause by testing/QA and not a user. Or you could use reflection to get the implementation name and try to generate a name... The problem with reflection is that it could effect sub classes and set up a code situation that would be difficult for a junior level developer to understand and maintain...
For that matter you could always generate the name from the class name thru reflection... Though in the long term this could be a nightmare to maintain... It would however reduce the amount of code needed on the implementations, which seems more important than any other concerns. Your could also use attributes here as well, but then you are adding code into the implementations that is equivalent in time/effort as a static constructor and still have the problem off what todo when the implementation does not include that information.