MYSQL: Stored function to replace category ids with names
Does this work? I wrote this up in a hurry on MariaDB so may not be 100% correct, but hoping it should work on MySQL 5.5 as well, or at least only require minor tweaks. It seems to work on your test data.

DELIMITER //
CREATE FUNCTION decode_path (
    `path_input` MEDIUMTEXT
)
RETURNS mediumtext CHARSET utf8
LANGUAGE SQL
NOT DETERMINISTIC
CONTAINS SQL
SQL SECURITY DEFINER
  BEGIN
    DECLARE cnt INT UNSIGNED DEFAULT 2;
    DECLARE cat_id INT UNSIGNED;
    DECLARE cat_name VARCHAR(200) DEFAULT '';
    DECLARE decoded_path MEDIUMTEXT;
    SET decoded_path = '';

    WHILE cnt &lt; LENGTH(path_input) - LENGTH(REPLACE(path_input, '/', '')) + 2 DO
      SET cat_id = SUBSTRING_INDEX(SUBSTRING_INDEX(path_input,'/',cnt), '/', -1);
      SELECT `name` INTO cat_name FROM category WHERE id = cat_id;
      SET decoded_path := CONCAT(decoded_path, '/', cat_name);
      SET cnt := cnt + 1;
    END WHILE;
    RETURN decoded_path;
  END
//
DELIMITER ;


Using this, I get:

SELECT path, `name`, decode_path(path) FROM category;
+--------------+-------------------+---------------------------------------------+
| path         | name              | decode_path(path)                           |
+--------------+-------------------+---------------------------------------------+
| /1           | A                 | /A                                          |
| /1/2         | B                 | /A/B                                        |
| /1/2/3       | C                 | /A/B/C                                      |
| /4           | D                 | /D                                          |
| /4/5         | E                 | /D/E                                        |
| /4/5/6       | F                 | /D/E/F                                      |
| /7           | G                 | /G                                          |
| /7/8         | H                 | /G/H                                        |
| /7/8/9       | I                 | /G/H/I                                      |
| /426         | Computers         | /Computers                                  |
| /426/427     | Other accessories | /Computers/Other accessories                |
| /426/427/428 | Laser printers    | /Computers/Other accessories/Laser printers |
+--------------+-------------------+---------------------------------------------+


So to store the result string from the function into a new column named path_long:

UPDATE category SET path_long = decode_path(path); 


Tested with MySQL 5.5 on db-fiddle.com here.