Oh dude, what ever you do, don't start by saying the common "Let's go around the room and each one of us will tell us about themselves". People hate that $hit, and it's an automatic buzz kill. I know what you mean, this is a bit of an annoying task, but you do have to make your team feel comfortable with you.
First find out if they are all comfortable with each other, that will give you a better idea of what you are up against, and you'd be half way home if they do. From past experience, I can tell you that if you bring silly games into the picture, afterwards it will be criticized as childish and rediculous and will be made fun of by the team. Not saying that all of the ideas from everyone else are stupid, just saying that the team may not appreciate them.
Our team leader took the group to Applebees (I was not there that day), but it seem to work. Not meening you have to order alchoholic drinks, but it's a relaxed atmosphere for everyone, and they can open up better.
Now, what did your boss mean by saying "I want you EACH to come up with an ice-breaker exercise"?
