template_preprocess_HOOK: what do I put for HOOK? My objective: get and change the breadcrumb within a custom module.

So it's possible that template_preprocess_HOOK is not even the best tool to be using for this, please mention that.

My question, though, is what HOOK to use? Should it be called template_preprocess_mymodule or template_preprocess_mytheme?
The HOOK corresponds to the theme function or template you are preprocessing.

If you wanted to alter page.tpl.php, you would use function MYTHEME_preprocess_page(&amp;$variables).

If you wanted to alter theme_item_list(), you would use function MYTHEME_preprocess_item_list(&amp;$variables).

A note, any dashes should be converted to underscores.

So views-view.tpl.php would be altered by function MYTHEME_preprocess_views_view(&amp;$variables).