The way I usually do this is to give each image a copy name (I use virtual copies for this) that reflects the intended image order. On the publish service I choose to rename based on the copy name.
Worth noting: the order of files outside Lightroom depends on how the device orders things (and no longer depends on Lightroom, once they've been pushed from the nest..).
That said, alphabetic ordering is the most common denominator - most devices are capable of displaying in alphabetic ordering.
Many devices will also display by date (usually capture date, but sometimes file-creation date..).
If you are publishing repeatedly the same photos, it get's very tricky indeed, since Lr will not re-order already exported files or take them into consideration. If you are re-exporting the whole lot, then it's much simpler.
Anyway, Lr5user-pt's method gives total control, but also has some drawbacks - you have to redo the copy names if/when you reorder photos (and then, re-export/publish all photos whose copy-names have changed).
TreeSyncPublisher will map in-Lr ordering to capture dates (in exported file metadata) or file-creation dates (by "touching" exported file dates on disk), so if devices support such ordering (by date), then published photos will remain ordered upon republishing, automatically. Granted, you have to be conscious of some stuff, and read the documentation, and learn to follow the rules..
(PS - a new version is coming out soon, which will have less "gotchas").
Exporder is like TreeSyncPublisher except it renames exported files to match in-Lr order - even trickier to use though, but works dandy if you're willing to learn where are the hoops to jump through..
The above mentioned plugins are free, and I wrote them - they only make sense if you will be re-ordering, or re-publishing changed photos.
If you are just exporting or publishing one-time/all-at-once, then the simplest method is just to prefix an ordering number to the export filenames:
Thanks Rob, since I just need to publish them once (hopefully), I used the sequence_original filename in the custom naming. I had tried this before, but I hadn't put an underscore between the sequence and the original file name. Since some of my filenames began with numbers, the sequencing didn't work when I sorted alphabetically. The underscore solved the problem.
Thanks, I thought of that, but I have over 600 images. I wanted to find a way to rename them automatically, and avoid doing it manually.