Well, to be honest, I’ve never really seem the abbreviation “MI”. I’ve seen the English one use though “IM” in a French context. The French Wikipedia page of Instant_messaging seems to agree with me

L’abréviation IM du sigle anglais Instant Messaging est parfois utilisée, également IMP pour « Instant Messaging and Presence »

But normally the expression is written completely or people, here in Québec, use a synonym like “clavardage” or “tchat”.

If an abbreviation is actually used, it’ll usually be the (if any) one for the program the person is using: MSN, AIM, ICQ (do people still use that??) etc.

I am french and I can say that there is no way to say that. We oftenly use the acronym of the website / software that is used but not Instant Messaging. By the way, I have never heard “clavardage”

As a French, I can’t see any common abbreviation for IM. I don’t know about Quebec, but “MI” is certainly not used in France or Belgium.

When not going for the long “messagerie instantanée”, people will more conveniently use the name of the service if there is one – “Skype”, “Communicator”, etc. “Chat” is used as well but in my experience, it tends to describe a multi-participant room more than a one-on-one instant conversation.

To go further than guillaume31, since chat is usually used when there are several people talking in the same room, one might use MP for “message privé” (equivalent of PM) for a one-to-one alternative.

It’s a bit more general though and doesn’t quite reflect the instantannée part, for which we have no equivalent (of IM I mean) in French.