AntiVigilate suggested:
"do bit - at this small size it's no longer a question of value but principles - what will you do in the world with a tiny little influence?"

I really like this suggestion, it indicates and celebrates the fact that the Bitcoin community is an open source community that relies on the contributions and efforts of its members. Its in many ways a "Do-ocracy".

One of the more recent criticisms of Bitcoin is that it is a scam that only benefits only the miners and initial users.   This will turn out to be the case if the community of people using Bitcoin don't continue "Doing" thing --writing shopping cart software, making web applications that allow for transacting, trading, managing, and using Bitcoin, preferably in an open source manner.  Other ways to contribute include advertising Bitcoin, learning about it and explaining it to others,  working to defend Bitcoin's legality, making it easy to generate tax records as needed for the use of Bitcoin,  and keeping Bitcoin's image and value as something that is useful for a wide range of transactions strong.     I'm not too worried about finding a few people who will use Bitcoin, but I am concerned that there could be a general consensus by many people that Bitcoin is a "Bad Thing" and shouldn't be used at all.

I think its important for people to remember that with using Bitcoin comes a responsibility for sharing the effort of building out many legitimate uses for Bitcoin and allow it  'To give us control over our finances by establishing a stable, secure, global, "democratic" currency.' to quote Gavin.  This won't happen without a lot of "do"-ing by many people who want to be Bitcoin succeed.  To that end, I think that to name 0.001 of a Bitcoin a "Do Bit" and even publicize it as to why we (might) choose this name, to serve as a reminder of service for the collective good, as seen by the many individuals in this Open Source project would mean a lot and look good for this effort.

People are naturally going to ask for a long time where whatever the name for 0.001 Bitcent ends up coming from. While we could just use a name based on existing scientific naming conventions, if we take an opportunity to add more cultural value to the name, we can make the social aspects of Bitcoins stronger.

I know that I'm being over optimistic that we here in this forum might actually be able to come up with it - Thanks for the point about the Network Effect, ShadowOfHarbringer.  But if we had a chance to set it, and have the back story about why many people started calling 0.001 bitcoins "Do Bits", that is, to encourage users to "Do things of value to the community of Bitcoin users, based on principles of OSS & Do-Ocracy", I think we'll be able to put off most of those who see Bitcoin as a scam.  I don't want people to "Do" blindly, or just because they'll get a good investment out of Bitcoin, I want them to "Do" because they see the value of an open source based financial system and the gifts of transparency, ease of transaction and more that it offers.

Perhaps there is another name which does an even better job of summing up the culture and values of the Bitcoin community other than Do Bit (DoBit)--I'd like to hear about it. 

Sorry if my idealism is uncomfortable, I'd be interested to hear what others think.  Thanks for the great suggestion, Anti Vigilante!