There are no such items in my family, so I have no clues from experience. If I did have one, I would give it to an archive if they were interested in it, where it can be preserved with appropriate temperature controls, with a proviso that it be made accessible to descendants or family's researchers.
To me, daughters make sense precisely because women typically lose their surnames on marriage. With a family Bible, you retain a record of women's lines which would be very difficult to duplicate through research.
However, nowadays records are much easier to trace, due to civil registration etc., so I think it should go to the eldest female who is actually interested in its contents and in sharing the information in it. Failing that, a male.