- If you are genuinely seriously interested in mentoring undergraduates in REU projects, then you would naturally be fantasizing about projects you might do. If you aren't all that interested in it, you can't fake it just by thinking up some projects. Note that there is some middle ground between 'genuinely seriously interested' and 'not all that interested'. Thinking up projects does show that you are interested in it. The ones you think up don't need to be great; they just need to be not delusional. Keep in mind that weaker schools will only have someone who is potential grad student material only once every several years, so those schools, if they are interested in undergraduate research, will want projects that someone who is not grad student material can tackle. (As a rough and not particularly accurate guide, by 'weaker school' I mean median SAT of incoming freshman below 1150 or so for a small liberal arts college; larger universities may have some stronger students even if the average is somewhat lower.) People in applied and computational areas have a huge advantage here because they can propose research that does not involve proving any theorems.
The truth is that you'll probably have a failure or two before you have a success, and it's not a big deal unless it is a total failure where the student gets nothing out of it. So don't worry too much about the specifics of the project; the point of having one is to show you are serious and not completely delusional (given the school and its students) about it.
- Unless the advertisement does not ask for a research statement or asks for undergraduate research to be addressed elsewhere, it goes in a section of your research statement.
Keep in mind that many schools that care seriously about undergraduate research will care only a little about your research except inasmuch as it generates projects for undergraduates. When applying for positions at such schools, you may need to reshape your research statement to say only generalities about your own research (since no one there will understand most of it anyway) and focus on the accessible parts and the undergraduate research. It may also be the case that such a school will not ask for or read a research statement, in which case you need to be prepared to put the material as a section in your teaching statement.
- It absolutely does NOT have to be in your primary research area, as long as it is an area where you have enough of an idea what is going on to actually guide a project. (In other words, you should know enough that having the project turn out to be a fairly well-known solved problem is not at all a risk.) In fact, many professors at undergraduate institutions have gradually migrated out of their dissertation areas to research areas that are more friendly to their students.