I'm resurrecting this thread to let you all know that I am about a week away from releasing exactly what was asked for here.
It is a component as well as CB plugin. It is J1.5 and CB 1.2 native (though I believe it'll work on the current CB as well - and I'll check that before release).
It allows users to invite others to your site. It can disallow public registrations without invite codes. It tracks who invites whom, how many invites users get total or outstanding, and such.
It integrates with the CB Subscriptions system (I am a beta tester) but does not require it. But if you have it (when it's release), it'll let you place new users in a subscription based on the invite code.
In addition to letting users invite others, it will also let you set "site invite codes" that you can publish in general, and also let you specify that they can only be used a certain number of times.
There's a lot more, of course, but those are the key features.
I'm less than a week from being code complete, so any last-minute feature requests aren't out of the question, but they'd need to be seriously cool at this point for version 1.0