Hubo algunos errores de sintaxis -
Uno de ellos es - todas las declaraciones deben estar al principio de la cláusula BEGIN...END .
DELIMITER $
DROP FUNCTION IF EXISTS get_string_class$
CREATE FUNCTION get_string_class(mystring VARCHAR(15))
RETURNS VARCHAR(15)
READS SQL DATA
BEGIN
DECLARE i INT;
DECLARE segment VARCHAR(15);
DECLARE String_Class VARCHAR(15);
DECLARE mystringlength INT;
SET i = 2;
SET mystringlength = LENGTH(mystring);
SET String_Class = NULL;
WHILE i <= mystringlength DO
SET segment = LEFT(mystring, i);
SET String_Class = (SELECT String_Class FROM string_class_list WHERE String_Begins = segment);
IF (SELECT FOUND_ROWS()) = 1 THEN
RETURN String_Class;
END IF;
SET i = i + 1;
END WHILE;
RETURN String_Class;
END$
DELIMITER ;