Modifying a module's function from another module A particular module has a theme function (theme_nice_menus_build()) that I'd like to give a slightly different output; I'd rather not modify the .module file to be able to continue updating the module when new versions are available.

Is it possible to modify that function from another custom module or from template.php?
All theme functions can be overwritten in template.php by replacing theme with phptemplate.
so your function in template.php should be like

phptemplate_nice_menu_build