This is something that can be done through MySQL alone.
Setup a unique key for the event column by running the following MySQL Command on your table:
CREATE UNIQUE INDEX `i_event` ON `TABLE_NAME_GOES_HERE` (`event`);
For more information: http://dev.mysql.com/doc/refman/5.0/en/create-index.html
Do this for every possible table you expect to see in the $tbl_name variable.
Then, change your PHP Query:
$sql="INSERT IGNORE INTO $tbl_name SET date='$mydate' , event='$myevent'";
$result=mysql_query($sql);
For more information on INSERT IGNORE: http://dev.mysql.com/doc/refman/5.5/en/insert.html
INSERT IGNORE simply does as it states... it will try to insert the row unless it fails validation (in this case from an index that you declared HAS to be unique).