Seconding hide and seek. We play in the big zones, and especially if you're playing in places with lots of nooks and crannies like Raktika, it can actually take hours to find everyone if you have a decent sized group. We've used our FC discord to voice chat and send each other hint screenshots as we play. We've also done something like "hiding in plain sight" where we all turn off names and glamour different gear or NPC-looking gear and try to blend in crowded zones like Limsa.

Maps, mount farms... We sometimes list out all the glamours people wanted from older content and rotate through the old raids/trials together to farm glamours or mats.

I haven't had a chance to try this on a big scale but we've done this with smaller groups and it's been fun--picking out a totally new weird-looking glamour for each other's characters and taking screenshots in the strange outfits.

We also do holiday-related stuff like a secret santa gift exchange, diadem trips together...

I think one that is especially fun for me are house redecorating parties. Getting people together to redesign the house is quite fun if you have artistic types in your FC.