Change the name of files uploaded through the media uploader.

class-m-custom-uploads-filename.php

<?php
/**
 * Class M_Custom_Uploads_Filename
 */

class M_Custom_Uploads_Filename {

	/**
	 * M_Custom_Uploads_Filename constructor.
	 */
	public function __construct() {
		$this->hooks();
	}

	/**
	 * Add the hooks used by the class.
	 */
	public function hooks() {

		add_action( 'current_screen', array( $this, 'maybe_mark_upload' ) );

		add_filter( 'sanitize_file_name', array( $this, 'maybe_change_filename' ), 10, 2 );

	}

	/**
	 * Check if we are in the right screen and add a filter for the media upload requests.
	 *
	 * @param WP_Screen $screen The current screen object.
	 */
	public function maybe_mark_upload( $screen ) {

		// Check the screen id
		if ( isset( $screen->id ) && 'toplevel_page_m-admin-page' === $screen->id ) {

			add_filter( 'plupload_default_params', array( $this, 'add_uploads_parameter' ) );
		}

	}

	/**
	 * Add the custom parameter to the set of default upload parameters.
	 *
	 * @param array $parameters The default parameters
	 *
	 * @return array
	 */
	public function add_uploads_parameter( $parameters ) {

		$parameters['m_admin_page'] = 1;

		return $parameters;
	}

	/**
	 * Check if the custom paramter set above is set and change the filename accordingly.
	 *
	 * @param string $filename The processed filename after sanitize filename function was called
	 * @param string $filename_raw
	 *
	 * @return string
	 */
	public function maybe_change_filename( $filename, $filename_raw ) {

		if ( isset( $_POST['m_admin_page'] ) ) {
			$filename = self::random_image_name();
		}

		return $filename;

	}

	/**
	 * Just a function used to generate random image names.
	 *
	 * @return string
	 */
	public static function random_image_name() {

		$characters = '0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz';
		$result     = '';
		for ( $i = 0; $i < 5; $i ++ ) {
			$result .= $characters[ mt_rand( 0, 61 ) ];
		}

		$result .= '_' . substr( md5( time() ), - 5 );
		$result .= '.jpg';

		return $result;

	}
}

// Comment this out if you'd rather call it somewhere else.
new M_Custom_Uploads_Filename();