OK, it was easy to adjust Jim's script to have a user change his mailbox password and quota remotely without logging on to Cpanel.
Simply take the same script that creates the email account and change the line that starts with $in to (all on one line):
$in = &GET /frontend/USER_THEME/mail/dopasswdpop.html?email=$email&domain=$domain&password=$password1&a=$quota HTTP/1.0\r\nAuthorization: Basic $pass \r\n&;
Make sure your form's input names match the email, password and quota variables on this line.
Replace the USER_THEME with bluelagoon, default, iconic or whatever the theme for the account is and leave all the rest in the script as is.