I don't know how hard this would be to implement, but making it so the chat actually recorded the last 20 or so things said in chat would really be an improvement. I strongly dislike the chat as it is, it feels really awkward to have to ask what may or may not have been said for the 30 seconds you spent refreshing your web browser.
Additionally, if I am on a tablet(specifically an ipad, no issues on other tablets though), moving away from the tab disconnects me from chat for some reason. This is less significant, but I figured this was worth mentioning.