1. Good support for multiple active clients, including mobile. If I see a message on desktop, I shouldn't get notified on mobile. If I read on one client, when I reconnect on another, it should show me the convo starting where I left off. Etc etc.
2. Eternal history (ideally configurable by users/channels) with decent search.
3. A few basic bells-and-whistles over IRC: preview images, allow file uploading, snippets, emoticons, profile images for users.
4. Multiple "team" / server support in the client.
I had high hopes for http://shout-irc.com/, but it ended up falling short in a number of areas. Notibly, its plugin support is not very good, so it's quite hard to add the sorts of features that I mentioned.