How important is it for one on the job market to have thought about suitable REU projects? An REU is a "Research Experience for Undergraduates" and entails undergrads (usually after their junior year) coming for 8-10 weeks out of the summer to University X to learn about and work on a problem under the supervision of one of the faculty at X. They are often funded by the NSF and the students get a stipend. REUs seem to be very popular among small colleges because they want to be able to sell themselves as a place undergrads can do research even if there are no grad students around. As a grad student who will be on the job market in the not-too-distant future, I'm starting to wonder about how best to prepare and how to fit REUs into my job application material. I work in stable homotopy theory, and I have never come across an open problem that I think an undergrad could do meaningful work on (ignoring the small set of really amazing undergrads who frequent MO and would probably not end up at my hypothetical REU). If I did come across such a problem, I'd probably solve it myself rather than hang on to it as a potential REU problem.
A bit more background: I will be applying for a large number and variety of jobs because with the job market as it is, it seems prudent to cast a wide net. I'll be applying for post-docs in the US and in Europe, and I'll also be applying for tenure track jobs at schools which are more focused (but not entirely focused) on teaching. Long-term I hope to end up at a good liberal arts college, but in the short term I'll look for post-docs to strengthen my research profile. If I end up in a post-doc in Europe, I assume they will want me to focus on research and not running undergraduate research projects. Thus, this question is more focused on applications to colleges where teaching matters at least as much as research. 

(1) should I spend some time seeking out a suitable REU problem in case a job asks me for project ideas?

I have no idea how common it is to ask a job applicant this type of question, but it seems like the sort of thing a small teaching college might want to know. 

(2) Supposing that I did find a good problem and stopped myself from solving it, where would be the best place to display that in my job application material? 

I've read up on teaching statements (on MO and elsewhere), but it seems that is not really the place to discuss plans for REUs. Similarly, it shouldn't go in the research statement. Is this just something I bring to the interview and hope they ask about?

(3) Do REU ideas need to be in the same field as my PhD research or could I just think up a good problem in a different field (e.g. graph theory where the background is more reasonable for an undergrad)

I'm amazed this hasn't been asked on MO before. I hope it's suitable, since many grad students frequent this site and would I suspect care about questions they may be asked in the job process. If it has been asked, please forgive the duplication. All I could find were questions about the merits of research if you are an undergrad, where to find info on REU type experiences, and the pros/cons for publishing with undergrads if you're looking to get tenure.
 A: 1) 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.
2) 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.
3) 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.
A: As a hiring committee member at an R1 school, I have never seen a candidate include ideas for REU projects as part of a job application, so I can definitely say that this is not expected and it won't hurt you not to have it.  I can also say that there's an order of magnitude larger number of liberal-arts colleges in the US than there are REU programs, so it's hard to imagine that most places will have in mind that you're going to run REU programs while you're there.  In other words:  I don't think you need to feel this is a necessary part of your job application.
THAT SAID:
If you have mentored undergrads in research projects, I would certainly talk about this in your application; any school, from LAC to R1, will be happy to hear that you have some experience in this domain.  But if you've never done it before, I can't see that there would be much benefit to saying that you have ideas about doing it.  Exception:  if you're applying to schools that you KNOW have REUs, and if you think that's something you'd like to be involved in, I would certainly say so in your application.  Running one is a huge amount of work (or so it seems from watching my colleagues; I haven't done it) and extra hands on deck are always welcome.   
A: I know from my own experience, undergraduate research is very important to many schools. As a post-doc I mentored a couple of REU projects and it definitely helped me land my current job. 
To answer your first question, "Yes". If you are interviewing at a mid-sized or smaller school, there's a very good chance this topic will come up. It's worth putting some time into thinking about projects. 
As for your second question, when applying to schools with more emphasis on teaching, try to work your REU ideas (or at least your interest in REUs) into your teaching and research statements. I'm less sure about applications to R1 universities - Maybe just in your teaching statement?
As for the last question, definitely "No". It's great to have a project related directly to your own research, but not necessary. For me, my subfield is affine Lie algebras and vertex operator algebras (not very undergrad friendly), last spring I mentored a group looking at partial fraction decompositions in Euclidean domains (still algebra, but definitely not my own research). Also, keep in mind they are undergraduates not PhD students. The research does not have to be really groundbreaking. Working out details in a basic example (in say "stable homotopy theory") might make a good project. 
A: At least as a place-holder answer: yes, primarily-undergrad colleges care about REUs, if only because it is the style, and because undergrads have come to expect it. Some undergrad places are surprisingly delusional about the "research" that their REU programs supposedly support. Never mind.
Indeed, it is unreasonable/implausible to imagine that any but the most elite undergrads would do any sort of serious/genuine research. Don't say that directly to anyone who apparently expects you to do exactly the thing that is absurd. The point is that what is truly expected is not genuine research. The true expectation, however couched, is that something palatable and new-to-the-kids be given to them to work on. After all, the real benefit of REU stuff is simply the non-classroom-setting, the collaborative nature, the non-adversarial relationship of mentors and students...
No, it is unlikely that even a fairly precocious undergrad will be able to engage with, or benefit from, looking at problems that make sense to a PhD'd person. It's possible, but to the extent that undergrad places care about REU-type-stuff, it's not really whether what the kids do is genuine (and they can't tell, in any case) but whether the kids are happy.
Problems that you imagine could be addressed by a sub-novice are what you want. Scarcely matters what "field". Right, typically perceived-as-nearly-prerequisiteless things work best, if only because of the perceptions of the kids. The practical point that REUs mostly occur just after peoples' junior years, so that the vast majority have taken no genuine math course at all, indicates the scope of the difficulty.
As a hint about how to not be excessively cynical/jaded, we can not that we do not encourage teenagers to act as though they were octagenerians, even tho', yes, eventually they will be. Not even as 40-yr-olds, etc. Various developmental stages are inevitable, and/but few people imagine how naive/ignorant they are. No sense in having that be the purported point!
One candidate starter-project is variant-on-standard-cliche, where the person develops the "obvious" parallel to completely hackneyed fun-thing. If they need the whole 10 weeks to figure it out, that's fine, and, if by chance/luck they run through it fast, it will have been educational, and may have provoked them a little. At that point, if it comes, you have to start thinking about a worthwhile project. NB, it's not "problem", but "project". 
For myself: I did quite a number of REUs, with some quite-better-than-average students, some years ago, but it was frustrating... and I've not done any since. The frustration was primarily that they wouldn't genuinely listen to me, because their preconception was that admission to an REU vetted their own "genius"... So that, apparently, they surely were supposed to do whatever they thought best, seemingly because "old people" had no worthwhile advice to give, etc. The worst downside to this is the apparent belief that "the literature" really does explain things. I got tired of spending energy on people who so militantly ignored me. :)
But when one is younger, one can/should/must behave at a level of "idealism" that is insupportable later... since, otherwise, one may never have done so. :)
