Why do some Buddhists believe that they should not eat meat for 49 days after the death of a family member?

This is probably not a Buddhist tradition but rather a tradition of your kin (家族). I remember seeing similar rules in my family's copy of my kin's genealogy book (it's actually a series of 7 books). The genealogy book's main function is to record the names of family members up to tens of generations ahead. But it also contains a special section on kinship rules that should apply to members of all families in the kin. Wikipedia has a brief introduction to Chinese Genealogy Books: http://en.wikipedia.org/wiki/Genealogy_book#Chinese_genealogical_books

I'm not sure of this tradition, but there is  specific reason for the number of days. In Buddhism, after a person dies, the mental continuum is said to enter an intermediate state which usually lasts from anything between 0-49 days. After 49 days the intermediate being will definitely have taken rebirth in one of the 6 forms of existence. While in the intermediate state, the being may still have strong attachments to its previous birth and thus special care is taken during these days which will allow the being to take rebirth again without feelings of attachment. Usually there are prayer services performed during this period, however abstinence from meat also might be a practice that might have been prescribed for some and become part of a culture.