I tried to click on your link and got nowhere, sorry, but maybe you can tell us how your home page search box is added? Via a widget or a PHP snippet in your template file? Or by a plugin, maybe the FacetWP plugin?
If it’s the plugin you’ll need to post in the FacetWP support forum (or the forum of whatever plugin is adding it).
If it is the stock WP widget, OR if it’s called in your homepage template file like this:
<?php get_search_form(); ?>
Then it uses searchform.php. If your Theme/Child theme doesn’t HAVE a file called searchform.php, then WP will use a default one of it’s own (from the core files).
(If it’s some other method, please be specific about it when you post back.)
NOW while you could, theoretically, create or edit a searchform.php file, keep in mind that this file would be called from *anywhere* on your site where you have a search box, which may include other pages and your 404 page.
SO if you modify searchform.php to change the value of that button, it will be changed everywhere.
If you have NO other search forms (be sure to check your 404 template as well) and you don’t care if the button shows the same value everywhere, then just modify searchform.php.
Open it in a text editor, it will likely look like this:
<form method="get" id="searchform" action="<?php bloginfo('home'); ?>/">
<input type="text" value="<?php echo attribute_escape($s); ?>" name="s" id="s" />
<input type="submit" id="searchsubmit" value="Search" /><br />
</form>
See that last line, where it says ‘value=”Search” ? Just change that value to your call to action.
If you don’t want that call to action to show up on every search form’s submit button, then just create your own search form for your home page only.
If it’s put on your home page via a search widget, just replace that search widget with a text widget and past in the form code BUT note that you’ll likely need the PHP Text Widget plugin, which lets you put PHP into text widgets.
If it’s added to your homepage via a template file, just replace the get_search_form call with your HTML code for the form.