create another table, called: expects
take into that the username's id and the news's id if he/she don't want to see anymore.
And this is a query to SELECT those news what is in the expect table with the signed in User:
<?
session_start();
$ui=$_SESSION["USERID"];
$sql="SELECT new,(SELECT COUNT(*) FROM expect WHERE expect.n_id=n.n_id AND expect.u_id=".$ui.") AS how_many FROM `n` HAVING (how_many=0)";
if(mysql_num_rows($result))
{
$result=mysql_query($sql);
while($row=mysql_fetch_assoc($result))
{
print "HERE COMES THE news list, with the link to the details...<br>";
}
}
else
print "You have not new news...";
//....
?>