This is so random, but I just started to think about what kids in different factions play with/do to spend time, and here's my thoughs:
in Abnegation they probably don't have much toys, maybe just skipping ropes or something else you do with others and have to take turns in. Anything too loud or rough is prohibited
in Amity they play with hand-made dolls and stuffed animals and the like, and they also get to dance and sing and play instruments, draw and paint, and do crafts like crocheting, knitting and sewing. Cooking for fun is likely something they do as well. In Amity loudness, running, etc. are fine, but playfights are not allowed
in Candor board games, card games, and such are a big thing, probably also truth-or-dare (but everyone always pics truth)
in Dauntless it's toy guns, cars, trains, etc. and rough-and-tumble play, playing cops and robbers (the tag game) and climbing everywhere, etc. They dare each other to do risky stuff all the time. And the play is basically always loud. Unlike the other factions, in Dauntless kids also get to play with make-up
in Erudite they get Legos and other building sets and chemistry sets and the like, puzzles and memory games are probably also a thing, and everyone of course reads lots of books. Oh, and unlike the rest of the factions, Erudite kids have access to smart devices
I also believe that telling stories is something that happens in every faction, but they're vastly different in what they're about and what the point in telling them is (e.g. in Amity, it's about making others laugh with something nice, while in Erudite you try to notice the illogicalities in the story and that's the fun part). Oh, and kids in every faction (well, maybe except Erudite) play tag, just in slightly different ways. And playing pretend is of course an interfactional type of play, too
