On Submitting to the Journal of Applied Mathematics Letters
(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.