Take the HPC REU as a good example (obviously you won't be able to do everything that they can do as an REU Site, but there are many things you can try to do). It was my top choice for REUs when I was a junior undergrad even though I never heard of the school. When undergraduates read the website they know exactly what they're going to do. For a student who doesn't know much about the subjects, showing a list of projects makes them interested even if they do not understand all of the details.
Also, invest some time in training. You are saying that you cannot find "eligible students at our own institution who are both qualified and interested". Do a bootcamp in what they need to know. Most of the students who attended the HPC REU knew how to program but didn't know anything about high-performance computing, so the detailed schedule showed us that we would learn this (and get credit). Most of the time undergraduates take "standard" classes so they will never learn these research skills even if that's what they want to be researching, so showing them that your REU is the jumpstart to their research career in the particular field generates the interest. Also, if you can show that this is applicable to them even if they are not into research, then you have a large crowd. If you look at other REUs they also have a few training weeks in some useful subjects as well.
Another thing is to try and make it a group project. Undergraduates haven't had enough time to learn subjects to "research-level" yet, and if it's an interdisciplinary project, it's almost impossible. Many mathematical biology projects take a group of students where one knows math, one can program well, and another may just be a straight biologist. They work together to solve the problem (or get as far as possible) and learn from each other.
Although I found my REUs via their websites, I know others that found out about their REUs via departmental emails. Send emails out to friends/administrators at other institutions to forward to their administrators. Some of the best places for recruitment can be liberal arts colleges because sometimes the department may not have every subfield. In a liberal arts college there are close ties between students and faculty and so many times the faculty will know about subfields that students are wanting to try out. Then the faculty may even forward the email to some of their students saying "I know you're interested in ______, check this out!".
Also, be ready to adapt what you're doing to the undergraduates. Most REU projects don't result in a publication, so if you're looking for undergraduates who will help you on your current project, it's probably a waste of time. Of course you don't want to go too far out of your expertise, but play around with ideas and try some new things. For example, if you're an experimental scientist who's interested in trying machine learning for your type of data, pair up with a new faculty member who does machine learning and get a group of undergraduates to try doing some machine learning on your data under joint supervision. Again, it's most likely not going to end up with results, but this new collaboration can turn out to be an exciting result itself.
Lastly, think about being the undergraduate. What is the undergraduate looking for? First and foremost, the best undergraduates are looking for what will look good on a graduate school application. Cater to that. Make the summer bootcamp worth credit. Guarantee that at least a technical report will be online. Teach them a few programming languages. Give access to GRE classes. Etc.