How about just a Bit?
EDIT: to also propose the suggestion Bbit (pronounce "bee bit") and to second the suggestion of bitbit.
And just a random suggestion, perhaps the new unit name should belong to the 0.0001 denomination (cent of a cent). We'll go through fewer names that way, less confusion. The most useful significant digits will all remain near the decimal point (four to the left, four to the right of it), it could be the only new name we ever need.
I am not sure that SI patterns or prefixes are the way to go. I occasionally use SI in regular speech just to be geeky (example, I like to say 5 "kilodollars" instead of 5 "grand"), and people look at me like I'm nuts and often don't even know what I mean, even though it should be obvious. No one seems to expect to see money measured like kilograms or millimeters.
Beyond the currently accepted "satoshi", I don't feel fond of honorary names ("gav" etc.), these seem like they could be a turnoff.
man i like bit