From what I've seen, people may want something different from hats...
Here's some (kinda strange) ideas I've had for a while...
A network/meta scavenger hunt to find a secret webpage for a prize?
Inspired by people sharing their knitting artwork (my personal favourite thing about the Bash), a very small Mario-Paint like composer. (I know it sounds a bit strange...)
A snowman builder/igloo decorating or something like that. (still inspired by knitting)
Remix on the first idea: various hidden icons (like a snowflake or something I dunno) hidden throughout sites and you can click on them and they get added to like a collection and there can be a leaderboard to see how many people have found
If I get any more ideas I'll add to this answer.