I think it is great when graduate students come up with problems to work on by themselves. There is probably no better preparation for a research career. Unfortunately, grad students in math only have about 3-4 years to learn about a field, produce new results, publish their research and write a thesis. This does not leave a lot of time for exploration. The role of the advisor is to try help select the most fruitful directions. Here are some of the questions that I found helpful:
1) Is the problem new, or has it been answered in some form before?
This is essential since you don’t want to have a student writing a thesis or a paper on something that is already known – and if it is known, it is good to find how others did it. This can then naturally lead to further questions.
2) Is the question interesting?
This is harder. One way is to have students think about how they could convince other graduate students that this problem is worth studying. After they can think about how to convince non-scientists.
3) Is it the question related to what we are doing in the lab/research group? Can it be answered in a reasonable time?
This is a more practical point. More senior graduate students that already have the majority of the thesis written, can have more leeway.
I am sure there are a number of other criteria here. In his TED talk, Uri Alon says that he approaches research like improv theater. This is great once a question is clearly articulated, and the group is looking for a way forward. Picking the right problem to work on is tricky. What other questions can help guide graduate students to a good problem, while encouraging them to take a role in designing their own research?