An unspecified short amount of time (usually less than an hour).
Can be used to indicate that events will follow a normal timescale, or otherwise be used to postpone an event for a time.
A 'bit' usually refers to the expected or average amount of time that should be taken to complete the task in question.
When postponing an event, however, a 'bit' can be considerably longer.
Eg, If you are due to visit a friend who lives a 15 minute drive away, to be round 'in a bit' could indicate that you will be there in about 1/2 hour, 15 minutes preparation to leave and 15 minutes driving.
The expression 'in a bit' is often used in Midland dialect as a farewell, similar to laters
'A bit' should not however be confused with 'Quite a bit', which always indicates an unusually large amount of time (or something else), and is analogous to a 'while'.