Hey, I'm the founder on SN. Thanks for the mention! You generated some extra traffic today.
There actually isn't a cost to using Lightning Login (lnurl-auth). It operates without really touching the Lightning Network, i.e. a channel. It just uses your Lightning wallet's key pair to authenticate you.
More on login: through the QR code (or lnurl string), I effectively send your wallet a randomly generated string, a challenge, it signs it using a deterministically derived key pair using my domain and your wallet key pair, then hits an endpoint on my server with the signature which authenticates you.