Try something like:
function nextEvent($dbname, $username, $nextevent) {
global $link;
if(!$link) die("Couldn't connect to MySQL");
mysql_select_db($dbname, $link) or die("Couldn't open Members: ".mysql_error());
$result = "UPDATE members SET nextevent=1 WHERE username='$username'";
mysql_query($result);
}
Which would set the nextevent to 1 where the username='some user name'. Or... if you're trying to set the username based on the next event, your query should like:
"UPDATE members SET username='$username' WHERE nextevent=1";
In other words, your query is all buggered up.