Rarest lost video games?

It's very hard to determine what exact video game is the rarest and is still considered "lost" by the wiki's standards. I honestly don't know how to answer your question, but I would say Birthday Mania for the Atari 2600 would be probably the rarest for me at least. While some people are known to have a physical copy of the game (and has been sold online a few times), none of them have dumped it to the internet for other people to enjoy.

Like I said before, it's a tough question with no real answer. 

Probably some of the ones with the biggest rarity factor are Mean Girls DS (which was cancelled at the last minute and we have no clue what stage in development it was at), Machot Drives it All/The First hardcore erotic Game in the world (for which we only got confirmation that it exists a couple months ago), the first Sonic the Hedgehog prototype (which even Yuji Naka doesn't know where it is), and Yeah Yeah Beebiss (assuming it exists as its own game and wasn't a typo or mistranslation for a game that is not lost). But like the other posters said, so many lost video games were simply cancelled or online/mobile games that just disappeared when they were taken offline. I'd suggest looking through the "Lost Video Game" category on the wiki and see if any of them catch your eye if you want to look for them. Also check if there are any threads on here, since there might be leads and search parties already that you can join in on.

Wait... Have it or had it?

Absolutely.

Agreed. That will be one video game mystery solved.

Though we have evidence of it existing, the Nintendo World Championships cartridge for the NES is the most rare video game of all time in my opinion. Very little people have it and is considered extremely valuable among collectors.