On Submitting to the Journal of Applied Mathematics Letters On their "guide for authors," they state the following regarding referees:


  (1) Please prepare a list of 8 (eight) proposed reviewers. You MUST
  include Full Names, Department, University, Country and Email
  Addresses for each proposed reviewer. It should be a globally
  geographically diverse list of potential reviewers; there should be no
  more than two suggested referees from any particular
  area/region/country. If you, or any of your co-authors, have submitted
  to our offices previously, please note that should not include
  reviewers on lists for your manuscripts in the last 2 years. All
  proposed reviewers MUST be fluent in English to ensure the integrity
  of each review and the correct processing of all manuscripts.
  
  (2) Proposed reviewers MUST BE experienced, well published researchers
  who are knowledgeable in your area of research. The editor-in-chief
  depends heavily on your suggestions for the choice of reviewers and a
  list of weak reviewers can be the basis for rejecting a submission.
  (Also minimize the number of leaders in the field who are typically
  too busy to be reviewers.) In addition, proposed reviewers MUST NOT BE
  former co-authors, instructors, co-workers, advisors, students, nor
  have had any other personal/working/professional relationship with you
  or any of your co-authors.


Part of what drew me to the journal was the short article size and the quick publishing time, however I fail to see (1) how this process can allow for expedient publishing and (2) how to gather said referees. I am supposed to gather a list of referees from around the world that I do not know while also somehow making sure they are "not too busy" to act as a referee. 

Is there any coherent way to accomplish a process such as this? How should I go about gathering said referees?
(I'm expanding my comment into an answer.)

First of all, I think this journal is asking for too much. Asking for eight recommended reviewers is unusual in itself (I have heard of journals requiring 3-5 names, but in my field it's usually not mandatory), yet this journal is doing just that and making a lot of additional hard-to-meet demands. For example, in a narrow field it's going to be near impossible to find eight researchers that are both knowledgeable in said area, and who haven't had any professional relationship with you or your coauthors...

When used well, the practice of suggesting reviewers can help expand the reviewer pool, or clarify to editors what their reviewers are knowledgeable about. When used this way, however, it seems more like a weird test, or maybe a bad sign that the editors don't have any contacts themselves. But hey, I guess if you want to submit your work there, better play ball. 


  I fail to see (1) how this process can allow for expedient publishing


In theory: The journal has a limited pool of people who have refereed for them in the past, and they might be mostly booked already, so the journal starts asking authors to suggest more potential reviewers. In the best case scenario the author suggests n new, great names and they can ask 1-3 of them to review the paper. The more possible referees, the higher the chance that some of them can write a quick report, which provides the expedient publishing you're after. However, the big downside is that if editors trust the suggested names too much, the peer review process gets ripe for abuse. I think that's what they're trying to address somewhat with the additional requirements.


  (2) how to gather said referees


Usually I have a take a more "free form" approach to this, but here's a potential algorithm:


Normally there will be a couple of people you just know would be interested, and be good reviewers. Start writing their names down in a list.
Go over the bibliography (mostly for the current paper, but possibly similar older ones) and pick out people you think could provide a constructive and in-depth critique. Write their names down in the same list.
Check the references highlighted in step 2 again, and ask yourself if you could include a coauthor of the person noticed in step 2. (This mostly is a way to select more early-career academics rather than the more obvious names of the leaders in the field. )
Think outside the box of the current paper. Do you know of any other people who have studied similar problems, or applied the same methods elsewhere? Who wrote books and papers you learned from, but didn't use directly in this work? If they seem suitable, write their names down too.
Next, go over the list, and try to come up with a motivation for each name. Remove names if they don't satisfy whatever requirements you might have. Also, in steps 1-4 you might have added authors that collaborate closely with each other. Try to reduce that if possible.
Ask your advisor, coauthors, mentors, ... for advice.
Repeat as necessary.


Now, my impression is that mathematics papers usually have shorter bibliographies than physics papers, and that there are fewer coauthored papers, so... good luck I suppose.