"bit" will never catch on for this unit, without context it is not specific enough to be clear what is meant by "bit". so it becomes useless to have this unit and still often be providing the context that we are talking about sub-units of BITCOIN.

However, for example "satoshi" and "finney" are specific enough that can be used without the context (only context is something countable) and everybody knows what they are.

Furthermore, if

1 XBT = 10,000 finneys
and
1 finney = 10,000 satoshis

and both sub-units are derived from people names
...it's perfect.

It's obvious, it's going to be finney.