I've come up with a working solution but remain open to other options. First, some rationale behind the original approach...
Because WCF uses thread pooling, anything based on a per-thread model may (and will) have a lifetime that extends beyond an individual request. I needed a way to store client context information pulled from the HTTP headers for each request as the information will be different each time. This means I can't persist the context information per-thread because the thread will be re-used.
Or can I?
The flaw in my logic was that thread re-use was the problem. In reality, each thread is only every servicing a single request at one time thereby making any information in that thread isolated to that request. Therefore, all I need to do is make sure that the information is relavent to that request and my problem is solved.
My solution was to refactor the Current property to reference a private static field marked with the [ThreadStatic()] attribute, ensuring that each instance was specific to the thread. Then, in my DelegatingHandler, which executes for each request, I reset the properties of the object for that request. Subsequent calls to Current during that request return the request-specific information and the next request handled by the thread gets updated in the DelegatingHandler so as far as my other code is concerned, the context is per-request.
Not perfect, but it at least gets me up and running for the moment. As I said, I am open to other solutions.
UPDATE
Upon closer inspection, this solution is not working as there is no thread affinity between the DelegatingHandler and the service code that is making use of the context object. As a result, sometimes my call to retrieve the ThreadStatic object works as expected but on other occasions I get a new instance because the code is operating on a different thread than the handler.
So, disregard this solution. Back to the drawing board.
UPDATE TO MY UPDATE
After discussing my problem with Glenn Block, it turns out that it is just a matter of making sure the context is set on the same thread the request handler (the service) is executing. The solution is to use an HttpOperationHandler instead of a MessageHandler.
According to Glenn, message handlers operate asynchronously which means they could execute on a different thread from the request handler (service) so we should never do anything in a message handler that requires thread affinity. On the other hand, operation handlers run synchronously on the same thread as the request handler, therefore we can rely on thread affinity.
So, I simply moved my code from a MessageHandler to an HttpOperationHandler and have the desired results.
You can read a full explanation here: http://sonofpirate.blogspot.com/2011/11/modeling-client-context-in-wcf-web-api.html