how about
<?php
$mem_id = $_GET['mem_id'];
require_once('./mysql_connect.php');
$query = "SELECT picture1 FROM memorial WHERE mem_id = '$mem_id'";
$result = @mysql_query($query);
if (mysql_num_rows($result)) {
$row = @mysql_fetch_array($result, MYSQL_NUM);
$pic = $row[0];
echo "<img src=$pic>";
} else {
echo "No Photo on File";
}
?>
this is more direct, and doesn't waste a variable.
if this doesn't work, consider the possibility that there actually is a record for this mem_id in the table, but it's an empty string. To exclude this possibility, you will need to add an additional WHERE clause.
since you are only looking for one row, you might also consider adding LIMIT 1 to your query.