They should buy Opera and open source it :)
No idea if that makes any sense for Facebook from the business point of view, but it’d be really cool. And they definitely can afford it.
EDIT: Now that I think about it, Facebook getting their own browser makes a lot of sense. They have a real chance of pulling off the mythical Social Browser.
Opera already has an extensive email client which they could integrate with Facebook Messaging, they have widget system which would also fit nicely with Facebook apps. Opera was always about integrated experience, full functionality out of the box, and Facebook could take it to the next level.