mysql> SELECT user();
+----------------+
| user()         |
+----------------+
| root@localhost |
+----------------+
1 row in set (0.00 sec)

mysql> SELECT current_user();
+----------------+
| current_user() |
+----------------+
| root@localhost |
+----------------+
1 row in set (0.00 sec)

/* user=jynus */ INSERT INTO test values (1);

mysql> SET @user := 'jynus';
Query OK, 0 rows affected (0.01 sec)

mysql> CREATE TABLE test (i int PRIMARY KEY);
Query OK, 0 rows affected (0.04 sec)

mysql> CREATE TABLE log (id serial, i int, user varchar(100));
Query OK, 0 rows affected (0.06 sec)

mysql> CREATE TRIGGER test_AI AFTER INSERT ON test FOR EACH ROW
       INSERT INTO log (i, user) VALUES (NEW.i, @user);
Query OK, 0 rows affected (0.02 sec)

mysql> INSERT INTO test VALUES (5);
Query OK, 1 row affected (0.00 sec)

mysql> SELECT * FROM log;
+----+------+-------+
| id | i    | user  |
+----+------+-------+
|  1 |    5 | jynus |
+----+------+-------+
1 row in set (0.00 sec)