I joined the chat at
dba.stackexchange.com and was quite surprised to find it was pretty active (at least during work hours, which is usually when I'm around)
I soon figured out that was because the users would spend time browsing SO questions looking for ones they'd like migrated to their site, discuss the question and its answers in chat, and if they liked it they would flag the question so a moderator would move it to their site.
I'm not sure if any of our users do this already, but I do know I've seen some great questions on SO which would be better on P.SE. This not only attracts users who find the question via Google to the site, but also kept the chat active since people would talk about the question and its answers before deciding if they should flag it or not.
Of course, it wouldn't work without some moderator participation. I would not want to be flagging questions for our site unless a moderator had already approved the question.
But if you guys are ok with it, I would probably spend a few minutes a day searching for good SO questions for our site and bring them up in chat for discussion, and hope other users pick up the trend :)