My homework asks a question along the lines of "Suppose a fair six-sided die is rolled and a fair coin is flipped. List the members of the event 'the die lands on an even number.'"
I can only guess as to what this question is asking. What does "members" mean in this context? Many sources use this term but it's remarkably hard to find an actual definition!
The sample space is the set of all possible outcomes of the experiment. For situations like this one where the sample space is finite, an event is any set of outcomes, in other words a subset of the sample space. So you should list outcomes where the die lands on an even number. Note that these will also include coin flip results; for instance $(2,H)$ should be one of your relevant outcomes.