There is considerable significance in an eighth day.
B’rit milah takes place then, after the baby’s first week of life. There are other first weeks, notably the first week of marriage and (l’havdil!) the first week of mourning.
Maybe we should institute an eighth day ceremony in those cases too.
Imagine. On the eighth day of marriage the sheva b’rachot are over and the newly-wed couple are on their own… a good occasion for a quiet ceremony of commitment to each other and of affirmation of the principles they espoused amid the emotion of the day of the chuppah.
On the eighth day of mourning the shivah is completed and the bereaved family are on their own (of course family and friends promised all week, “We’ll always be there for you!”, but it does not always happen).
Here too there could be a quiet ceremony of dedication to each other, undertaking that the family will not let itself fall apart for any reason, financial or other; and determining that though they have to return to the world and go back to work and normal activity, the inspiration of the deceased will be with them for ever.