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.
The site however does use the Lightning Network to provide Sybil resistance. Upvotes/posts/comments cost 1 sat, ~1/2000 of a dollar. This helps gate the site allowing mostly Bitcoiners in and imports an incentive structure - those more like in person social interactions.
Rather than staking your reputation, you're staking Bitcoin. Rather than earning just reputation, you're earning Bitcoin (your earned karma/upvotes is Bitcoin you can withdrawal from SN at anytime).
You can authenticate yourself with a tiny tiny bitcoin payment over the lightning network.