IMHO, 'for a bit' is for longer than 'for a second'. They both mean 'for a short amount of time, unspecified', but when you say: 'I'm going away for a bit' - it may mean a few weeks or even months. While 'for a second' means really not longer than a few minutes. 'For a bit' is, imho, like 'for a while'. I'm curious what other people think. Jo.

For a bit means a short period, perhaps a few hours in this case. For a second means a very short period, perhaps long enough to make a very quick call, in this case.