I had the exact same need and I solved it with SiteAccess plugin plus a default instalation plus some small fixes to SiteAccess. 

First get the module:

http://community.elgg.org/pg/plugins/project/384964/developer/uddhavadasa/site-access-26-full-working-release-for-17

After you install, go to the plugin settings (from the plugins list page) and enable "invitation only". There are many bugs in the plugin, but you can ignore them if you use it for this purpose only.

One modification I had to do to get it to work was to anihilate one of the plugin functions by adding

return true;

immediately after    

function siteaccess_validate_captcha() {

in the start.php file of the siteaccess plugin.

If you plan to fix additional bugs to get the module working again on 1.8.1, I'd be happy to work with you on a new release.

Yeah, S/A can handle the simpler situation of 'invitation-registrations'.. But what Ronnie's asking for is " current members can invite new members to join.. " -- so that will need to tie-in with Invite Friends - to generate the invitation, etc and when a new user 'registers' -- need to check if hey were invited by another existing User and then process that accordingly.

I've not yet tested the invitatation of friends feature in Elgg 1.8 when the "allow registrations" option is turned off in site settings. Basically, you can configure Elgg 1.8 to only allow logged-in users to browse your site and you can also disable the normal account registration. Accounts can still be created by admins if the normal registration is turned off. If the registration of accounts for people invited by already existing members would still work, it wouldn't be necessary to code an additional plugin. Although in this case I would guess an additional site administration option would be best to define if invitations should be allowed when normal registrations are disabled. Maybe it would be worth to open a ticket at trac about this issue to inform the core developers about this issue and it might be included in a future Elgg release.