Hey Hazem, how have you been, my friend?
In the context concerned, the idiom ''a bit'' means the same as ''a brief amount of time'' / ''a moment'' / '' a short period of time''. So, by saying ''It may take me a bit to be convinced that they don’t want my advice'', you are quite implying that it may take you some time or even a short period of time until you are convinced that they don't want your advice.
Below you shall find some other useful examples for a better understanding:
E.g.: ''I should be over in five minutes, so I'll see you in a bit!''
E.g.: ''Please, just wait a bit, will you?''
E.g.: ''I can't quite remember the last time she was silent for a bit.''
E.g.: ''It will probably take her a bit to accept the truth.''
Hope that helps!