Some organisations are realising the value of saving instant messages just as we save emails. Many IM clients let you keep logs and Gmail chat conversations are saved alongside email conversations.
The odd tech out here seems to be SMS. It's instant and most of the time you'd want messages to be discarded for privacy. But what if you want to save messages? I've been known to type messages into LJ. Someone on the train right now is handwriting SMS messages into a notebook.
Phone manufacturers need to provide a way to save messages to a computer. (Not sure how many already offer this.) And phone companies could offer a service to save messages on a server for a period of time. This carries an increased privacy risk, so it should be an option. But the reality is that messages are no doubt already saved for a period for law enforcement reasons.
And most phones allow you to save the SMS (though you've got that whole privacy issue in the event someone picks up your phone and goes through the messages - much like if anyone logs onto your PC with your login and goes through your IM clients or inbox)
My next post I make on my LJ has more details and, once it's uploaded, will be here.