Okay first of all the question:
What is the game pin-finger and how or why would a spider want to play it?

No idea if it is another name for thumb wrestling or what. All I know is spiders don't have fingers.

Now games spiders could play:

Leg Wrestling: opponents lie on their backs opposite directions, entwine legs at the middle or largest joint, you need then push forward and flip the opponent over.

Cat's Cradle or in this case Spider's Cradle: Game of creating figures with string https://en.wikipedia.org/wiki/Cat's_cradlehttp://

Knot tying: take string and tie fancy knots in it that have different functions an practical uses

Puppeteer- Attach strings to a doll and make it move around

Web Slinging- Shoot targets at a distance with web shot, just like spiderman

Web Swinging- just like spiderman, maybe swing and let go to land on a target area.

Don't know if they be any good at skipping rope, but they could make the skipping rope and super sticky skipping ropes might be interesting.

So anyone else got interesting spider games? Don't worry about it being hard to implement as something in game, it could just be something described in text in a talk dialog.