All use cases should be written down somewhere. Depending on how important they are, the amount of detail varies -- from a fully-dressed use case to barely more detail than a descriptive name. Like others here suggested, the key in not getting stuck in over-specifying everything in the world is to give the "brief" treatment to everything, and then choose the most important UCs to elaborate on. Every iteration only touches the tip of the UC iceberg -- and this doesn't just pertain to implementation, but also to use case elaboration.
The trick in ranking which use cases to work through is to evaluate their importance -- but I have to disagree with some of the folks here who have suggested that the ranking should be based on the difficulty of writing Unit Tests, or on "did the customer ask for it."
Two factors that should be considered when choosing which use cases to elaborate and to work through first:
Feature importance. The customer can provide some ranking guidance.
Risk. X happens rarely, but when it does, patients get irradiated with lethal doses of radiation. That's probably really important even if it's not likely to happen. Customers often do not have enough information to evaluate risk, and pushing this onto them is not agile development, it's shirking responsibility.
It needs to be understood by every member of the group that not all use cases will be designed in detail in any given iterations -- just the top ones. Everything else is important, too -- and it will see the light of day next week.
Edit Having read your update to the question -- research labs have goals, too. It's easier to get sidetracked when there is no paying customer (other than NSF) watching what you are doing, but you still need to do the research, publish papers, present at conferences, discover something (or discover that the something doesn't work). So your question becomes not "did the customer ask for it?" but "is this critical to the problem we are trying to solve?"
For example: Him: 'we need to code resilience into the system, since if the power goes out, we have to spend an hour recovering otherwise." You: "how often does the power go out? Can we recover as things are right now, or do we risk completely losing all research for the past 6 months? Is our research on recovery and resilience, or on better fudging of pudgy widgets?" ...