I like the idea of making it clearer on the Invite Someone page how many invite codes actually are available, but I don't like the idea of putting the number in the header. Once a user has hit the point where everyone they know who wants an invite code has one, the number is likely to be permanently non-zero, and that could quickly get annoying.

Additionally, having a second number up there would make it more difficult to glance at it and quickly figure out the status of one's inbox.