I became an editor recently after being turned down about 5 times! Just be really honest about your own sites and don't list one of them as an example. Also take your time over choosing example sites!
You can use one of your own sites IF it fits the category that you are applying for. If yopu are honest and declare all your sites and you do not offer one when it fits exactly the category, what will the p[erson who looks at your application think, probably that you are nuts. Just don't give the impression that your reason for applying is to list your own site. It can be easier to apply for a regional site, say your home town, that way any sites you offer will be absolutely right because they will fit a locality. You can always learn the basics and then apply for other categroies. Chose a category with between 15 and 100 sites already listed INCLUDING any sub categories. Highly unlikely that a category outside these will be given to a new editor. Make sure your sites really do fit the category, ensure that you title the sites according to guidelenes, likewise the descriptions and ensure your spelling and grammar is spot on. As said above make sure you are honest about any affiliations.
Just provide what they are looking for: An honest person that can learn how to edit. Perfection is not needed first up, you just need to show that you have the potential Essentialy, that is why you are asked to suggest 3 websites, so they can see if you have a general idea about placement, titles and describing the sites. Good luck
Snooks has pretty much summed it up. It took me two attempts to get accepted. Just read all the available information on the DMOZ Site and also the resource zone and take care with your application. Also be sure to disclose all sites which you are affiliated with, including those that have forums which you moderate on.
Now hypothetically.......if it took me 6 attempts would anybody laugh? BUt at least i finally had success.
For Become DMOZ Editor - Be Honest - Expertise To Understand Suitable Category - Avoid Spammer Website