How is a variable passed to theme_hook (theme_item_list, FacetAPI)
It is not necessary to create a new theme function. You could use the existing theme_item_list and extend it with double underscore. Rename item_list_aggregate to item_list__aggregate in FacetapiWidgetLinks::execute. And implement theme_item_list and theme_item_list__aggregate in your theme.

Look at node_title_list in core, this function uses the same pattern.