Add filter by name to WP REST Posts End Point.

functions.php

<?php
add_action( 'rest_api_init', 'rest_api_filter_add_filters' );
 /**
  * Add the necessary filter to each post type
  **/
function rest_api_filter_add_filters() {
	foreach ( get_post_types( array( 'show_in_rest' => true ), 'objects' ) as $post_type ) {
		add_filter( 'rest_' . $post_type->name . '_query', 'rest_api_filter_add_filter_param', 10, 2 );
	}
}
/**
 * Add the filter parameter
 *
 * @param  array           $args    The query arguments.
 * @param  WP_REST_Request $request Full details about the request.
 * @return array $args.
 **/
function rest_api_filter_add_filter_param( $args, $request ) {
	// Bail out if no filter parameter is set.
	if ( empty( $request['filter'] ) || ! is_array( $request['filter'] ) ) {
		return $args;
	}
	$filter = $request['filter'];
	if ( isset( $filter['posts_per_page'] ) && ( (int) $filter['posts_per_page'] >= 1 && (int) $filter['posts_per_page'] <= 100 ) ) {
		$args['post_per_page'] = $filter['posts_per_page'];
	}
	global $wp;
	$vars = apply_filters( 'query_vars', $wp->public_query_vars );
	foreach ( $vars as $var ) {
		if ( isset( $filter[ $var ] ) ) {
			$args[ $var ] = $filter[ $var ];
		}
	}
	return $args;
}

nice work

How to filter with LIKE on product_name?