Chat has quite a bit of useable real estate, and I feel disconnected with the primary function of the site when I'm solely in chat. This is likely in large part due to a lack of a Global Inbox while in chat.
A few of my own ideas:
A private onebox could pop up in your current active room(s?) with question details, but instead of the question itself, some information about the change, or the change itself could fill the body text. These should be styled differently than the room is, in order to convey that this is a private onebox to you alone.
Or perhaps, there could be a global inbox in chat that acts as a personal, private room, and presents notifications in a large, chat style, display. And the button to access the inbox carries the same number notification badge on top of the button when a new unread event comes in.
If an entire private room is made for this function, then perhaps instead of starred/pinned messages, reputation graphs could be displayed there.
The sidebar could also carry recently active questions you've starred or something to that effect. Maybe most active? ("Top 5 Questions You've Starred").
