Millies, as a friendly form of millibitcoins.
I like it because it goes with Mikes, a friendly form of microbitcoins, and someone is sure to do a nice logo of Millie and Mike.
Millies have a nice sound to it. Mikes - I'm not so sure about
I would call 1/1000th a Millibit for business (semi-official of Millibitcoin) and Millies in casual dealings