... where lower(jmeno) like lower($param || '%')

mysql> select * from a;
+----+------+
| id | t    |
+----+------+
|  1 | AAA  |
|  2 | Aaa  |
|  3 | aaa  |
|  4 | bab  |
|  5 | baa  |
|  6 | baa_ |
|  7 | A_a  |
|  8 | A_A  |
+----+------+
8 rows in set (0.00 sec)

mysql> select * from a where t like 'A%';
+----+------+
| id | t    |
+----+------+
|  1 | AAA  |
|  2 | Aaa  |
|  3 | aaa  |
|  7 | A_a  |
|  8 | A_A  |
+----+------+
5 rows in set (0.00 sec)

mysql> select * from a where t like 'a%';
+----+------+
| id | t    |
+----+------+
|  1 | AAA  |
|  2 | Aaa  |
|  3 | aaa  |
|  7 | A_a  |
|  8 | A_A  |
+----+------+
5 rows in set (0.00 sec)

mysql> select * from a where t like 'a\_%';
+----+------+
| id | t    |
+----+------+
|  7 | A_a  |
|  8 | A_A  |
+----+------+
2 rows in set (0.00 sec)

mysql> select * from a where t like 'a\_A%';
+----+------+
| id | t    |
+----+------+
|  7 | A_a  |
|  8 | A_A  |
+----+------+
2 rows in set (0.00 sec)