Mr. HIT

Stand in a circle.  One person announces, "I am Mr. Hit!"  Mr. Hit begins walking directly (but slowly, at first) towards another person in the circle, with his hands out in front of him like a zombie.  If he touches (hits) the person, they are "out" and must leave the circle.  The only way the intended victim can stop Mr. Hit is to call out the name of another person in the circle before any contact is made.  No fair running away.
Once a name is called out, that person instantly becomes Mr. Hit and begins advancing on a victim.  (He doesn't have to announce that he is Mr. Hit after the first time.)  Again, the only way the victim can save himself is by calling out a name.
Continue play until all but two people have been eliminated.  It gets harder and harder, because as people are "out" the fund of names grows smaller.  You can't say the name of a person who is out--it must be someone who is still in the circle.  It sounds easy, but it isn't.  It is very difficult to think with Mr. Hit bearing down on you.
This is great, if frustrating, fun, and although, as I've said, I think it is great for any group, it is also hands down the most effective "name game" I know.  Something about the sheer panic you feel when Mr. Hit is coming at you and you can't think of a name to say really makes those names stick in your head.  Doesn't work with little kids.
Pointers Gradually move faster, but never really fast.  It's quite scary enough without that.
Some groups find that they have to change the name of the game to "Mr. Tap" to avoid injury.  You know your group best.  There's really no reason to do more than lightly touch the victim.