I usually suggest the student spend a couple weeks, or more if they want, taking a look at a little bit of material in my area and then come talk more with me, so that we can both get a sense of whether the area is a good fit for the student. Typically, I print out a survey paper in my area and suggest the student start reading part of it; and I also ask the student to try to do a couple of exercises from a book in my area, suggesting the student pay attention while doing this both to whether the material seems to mesh well with the student's talents and also to whether they find it enjoyable/engaging. I find it very informative to see how this goes and to chat with the student after they've grappled a little with this material; I tell them I don't think either one of us should make a definite decision until after they do this. Of course I try to make it clear that it's fine to come back with questions about the material. This also buys me a little time to learn more about the student -- like Lee Mosher mentioned e.g. how they did in coursework.

Some of the students who have approached me seemed to think combinatorics (my area) would be the easy route through grad school, which I do not believe is true at all -- so I try to communicate that it's an area where on the one hand questions may sometimes have simple statements, but on the other hand may require a lot of ingenuity to solve. Based on this experience with students, I think it's also important to make sure students aren't making decisions based on misconceptions. So I second everything Lee said too.

Others here have surely had many more students than I've had so far, but I did have the experience of being approached by a large number of students right after starting a tenure track job, so gave this a lot of thought then.