I would fire back a JSON response containing some sort of error code. To process it, you need to define fnServerData as you have surmised. However, I would strongly consider the use case before using the error callback for this reason:
Error is simply any problem fetching the resource, and uses status codes. Let's say the session is terminated on the server. User requests the data, the server sends back a 500 error. Error callback says, "Well, that sucked. Let's redirect to a login page." Everything works great.
However... the user makes a request. The request is taking a little bit of time for whatever reason (large data set, network conditions). In the meantime, the user decides to navigate to another page, interrupting the call-response loop. With no response, the error callback is tripped and the user is redirected to a login page (or whatever function is set in the error callback).
The problem is that there's no status code that I'm aware of (I'd love to know, though! I don't mind being wrong here!) for 'session has expired' in order to capture and process.
I'm not saying you "shouldn't" or "can't" use the error callback. But the function has to take into account errors other than the session expiring. You might need to get into processing differently based on status codes. If your function handles all those cases elegantly, great! In one application, we are indeed redirecting to login page, and the error callback would often trip due to a false positive, dumping the user incorrectly to the login page. For the case of "session expired" we are catching it in the success callback via a JSON message.
[updated after Dave's excellent comments: ] If you have your server return a useful server error (401, 403, 550 or whatever makes sense in your scenario) then using fnServerData with statusCode parameter in the .ajax() call (that's a mouthful!) would work just as well. I think it's the same amount of work: return JSON via a method you've already written, or return status error via methods you should already have access to. Choose which one makes sense for you.