For the ajax request to map to your module's helper the class must be named "modFilefinderHelper"


From the documentation:


> Module requests must include the module variable in the URL, paired
> with the name of the module (i.e. module=session for mod_session).
> 
> This value is also used for:
> 
> The name of the directory to check for the helper file, e.g.
> /modules/mod_session/helper.php The class name to call, e.g.
> modSessionHelper

[relevant doc](https://docs.joomla.org/Using_Joomla_Ajax_Interface)

---

Edit after question update

I've updated and commented two of the files where you're having problems

    <?php
    class modFileFinderHelper
    {
        function __construct() {
            $this->currentdirectory = basename(JPATH_BASE);
        }
    
        // static methods only (the constructor above wont run for your ajax request)
        public static function FileFinderAjax() {
            // Here we get the data passed via ajax - note the values themselves have not been filtered!
            $data = JFactory::getApplication()->input->post->get('data',[],'array');
    
            // Presumably you would here do something worthwhile
            $response = [
                'something' => 12345,
                'your_path' => $data['path'],
            ];
            // Usually you can dump json and die, but there's a inbuild responder that'll encode things and die on your behalf
            // you'll see client side it also adds a few generic values like "success"
            echo new JResponseJson($response);
        }
    }


    jQuery(function($) {
    
        // grab a dom selector so we don't have to grab it each time we update
        var filefinder_container = $('#filefinder_container');
    
        function loadContent(path) {
            console.log({path: path});
            var request = {
                option       : 'com_ajax',
                module       : 'filefinder',  // to target: mod_filefinder
                method       : 'FileFinder',  // to target: function FileFinderAjax in class modFileFinderHelper
                format       : 'raw',
                data         : {path: path}
            };
            console.log(request);
            $.ajax({
                method: 'POST',
                data: request
            })
            .success(function(response){
                console.log('response: '+response);
                // we've got json but we need an object so let's parse it!
                response = jQuery.parseJSON(response);
                // and now we can use it like this.
                filefinder_container.text(response.data.something + " and " + response.data.your_path);
            });
        }
        console.log('onload, calling ajax with default path value: '+$('#filefinder_container').html());
        loadContent($('#filefinder_container').html());
    
        $('#path').on('change', function() {
            console.log('change event triggered, calling ajax with path value: '+$(this).val());
            loadContent($(this).val());
        });
    });