Facebook – Join a Facebook group secretly

My friend left my group. How can I send a second invitation?
Unfortunately, you will not be able to send a second group invitation to the same friend. The person will have to request to join the group again.

That's from Facebook Help, so my best suggestion is to make it open (or even closed, since one can ask to join a closed group) and let this take effect before your friend starts searching it. I could be wrong, but my guess is he could not find/join it because of the fact that the change is not instant.

Facebook does not provide a way to have groups that anyone can join instantly. Someone already in the group must add them, or they can request approval from a group admin.

The likely reason for not providing this is that it would attract spammers. Old-style open groups had spam problems and they did not even have the new group notification or chat options. Unlike posting to a page, all group members are now notified by default of any new post to a group, and also for groups with no more than 250 members it is possible to send chat messages to all members of the group including people who are not confirmed friends.

If you want anyone to be able to join and don't care about the notification or group chat features, then you may want to use a page rather than a group.