I thought of the exact same thing for my site some time ago (but did nothing about it

).
Thought it would be cool to try to emulate a military "Mess" where members are by invite only. Since there's no way I could personally know all those "eligible" to be members, having a system were existing members can send invitations was the only way.
In the end I scrapped the idea because it seemed to pretentious (in my context, anway) but I think it has a lot of merit, especially for high volume sites.
I also like the "accountability" of know who invited who. Be neat to follow the threads of invitations...